To Make a Long Post Short:
According to various calculations made from TOS data, Pike would be somewhere between 19.247 years older than Spock and 22.312 years younger than Spock, but probably older than Spock. Comparing the birth dates of various actors who portrayed Pike and Spock together, Pike should have been 5, 13, or 21 years older than Spock.
A Long Exercise in "Time Computation"
For many reasons I consider the official chronology of Star Trek to be "merely official" and not canonical and partially inaccurate. Therefor I prefer to use only canonical information for Star Trek chronology.
Jeffrey hunter was born 25 November 1926 and Leonard Nimoy was born 26 March 1931, and so they were aged about 38 and 33 respectively when "The Cage" was filmed 27 November to 18 December, 1964.
After Captain Pike and Number One are captured separately by the Talosians, Spock is in command and so was already third in command by the time of "The Cage".
SPOCK: Address intercraft.
GARISON: Open, sir.
SPOCK: This is the acting captain speaking. We have no choice now but to consider the safety of this vessel and the remainder of the crew. We're leaving. All decks prepare for hyperdrive. Time warp factor.
TYLER: Mister Spock, the ship's controls have gone dead.
That scene is not in "The Menagerie, Part 2", but this is:
PITCAIRN: Sir, it just came on. We can't shut the power off.
SPOCK: Mister Spock here.
TYLER [OC]: All power has come on, Mister Spock. The helm is answering to control.
(first Colt, then Number One are beamed aboard)
So Spock seems to already be senior in rank to, for example, Lt. Tyler, portrayed by Peter Duryea who was born 14 July 1939 and was thus about 25 when "The Cage" was filmed.
Since Spock is clearly an alien in "The Cage" and nothing is said about the lifespan of his species, there is no way to decide if Spock is older or younger than Leonard Nimoy looks, or by how much.
In "The Menagerie" Spock says several times that the events of "The Cage" were "thirteen years" earlier.
SPOCK: This is thirteen years ago. The Enterprise and its commander, Captain Christopher Pike.
SPOCK: Talos controls the vessel now, sir, as they did thirteen years ago. You've asked me why. You'll see the answer now.
So "The Menagerie" must be between 13.00 and, say, 13.99 years after "The Cage". So if Pike actually was about 38 years old in "The Cage" he would be about 51 in "the Menagerie", and if Spock was about 33 in "The Cage" he would be about 46 in "The Menagerie".
Since Leonard Nimoy, born 26 March 1931, was about 35 when "The Menagerie" was filmed 11 to 18 October 1966, if Spock was about 35 in "The Menagerie" he would have been about 22 in "The Cage".
So one might assume that if Spock ages similarly to humans he should be about 22 to 33 in "The Cage" and about 35 to 46 in "The Menagerie".
"The Menagerie Part 1" has other important information:
MENDEZ: You ever met Chris Pike?
KIRK: When he was promoted to Fleet Captain.
MENDEZ: About your age. Big, handsome man, vital, active.
KIRK: I took over the Enterprise from him. Spock served with him for several years.
SPOCK: Eleven years, four months, five days.
If Spock began serving with Captain Pike at the exact beginning of "The Cage", which is unlikely, and served with Pike for eleven years, four months, and five days, or about 11.347 years, and if "The Cage" was about 13.000 to 13.999 years before "The Menagerie", Kirk became the commander of the Enterprise about 1.653 to 2.652 years before "The Menagerie". But if Spock began serving with Christopher Pike - on or off the Enterprise - some time before "The Cage", which seems likely, Kirk would have become commander of the Enterprise a corresponding span of time earlier, and thus possibly months or years earlier than 1.653 to 2.652 years before "The Menagerie".
In "Where No Man Has Gone Before":
KIRK: Have I ever mentioned you play a very irritating game of chess, Mister Spock?
SPOCK: Irritating? Ah, yes. One of your Earth emotions.
KIRK: Certain you don't know what irritation is?
SPOCK: The fact one of my ancestors married a human female
KIRK: Terrible having bad blood like that.
This shows that Spock is part human and thus if there is a difference in life spans his may between humans and that of his species.
In "The Corbomite Maneuver":
SPOCK: However, it was well played. I regret not having learned more about this Balok. In some manner he was reminiscent of my father.
SCOTT: Then may heaven have helped your mother.
SPOCK: Quite the contrary. She considered herself a very fortunate Earth woman.
This shows that Spock is half human and likely to have an intermediate lifespan. It also implies that Spock's mother is dead.
In "The Naked Time":
KIRK: Where have you been? What happened?
SPOCK: My mother. I could never tell her I loved her.
KIRK: We've got four minutes, maybe five.
SPOCK: An Earth woman, living on a planet where love, emotion, is bad taste.
KIRK: We've got to risk a full-power start. The engines were shut off. No time to regenerate them. Do you hear me? We've got to risk a full-power start!
SPOCK: I respected my father, our customs. I was ashamed of my Earth blood. (Kirk slaps him) Jim, when I feel friendship for you, I'm ashamed.
And this also implies that Spock's human mother is dead by the first season. If Spock is about 35 to 46 in "The Menagerie" and if his mother was about 20 to 40 years old when Spock was born, she should be about 55 to 86 in "The Menagerie", and that age range gives no certainty about the probability of her being alive then.
Then in "Journey to Babel" Spock's parents Sarek and Amanda are seen.
MCCOY: Mister Ambassador, I understand you had retired before this conference was called. Forgive my curiosity, but as a doctor, I'm interested in Vulcan physiology. Isn't it unusual for a Vulcan to retire at your age? After all, you're only a hundred and two.
SAREK: One hundred two point four three seven precisely, Doctor, measured in your years. I had other concerns.
Unfortunately Amanda's age is not stated, but she does look a bit young to be the mother of Spock. Amanda is portrayed by Jane Wyatt (born August 12, 1910) who was about 57 when "Journey to Babel was filmed 21-28 September 1967. Jane Wyatt was 20 years, 7 months, and 14 days older than Leonard Nimoy.
The script for "Journey to Babel" describes Amanda as 58 years old. If "Journey to Babel" was immediately after "The Menagerie" Spock might be about 35 to 46 in "Journey to Babel", making Amanda about 12 to 23 when Spock was born. Of course the time between "The Menagerie" and "Journey to Babel" could make Amanda's possible age range younger. Thus I can imagine the future equivalent of tabloids having headlines like "Alien Ambassador Elopes with Teenage Earth Girl" at the time of their marriage.
AMANDA: You don't understand the Vulcan way, Captain. It's logical. It's a better way than ours. But it's not easy. It has kept Spock and Sarek from speaking as father and son for eighteen years.
Sarek may have stopped speaking to Spock when Spock said he wanted to go to Starfleet Academy, when Spock applied, when Spock was accepted, when Spock entered, when Spock graduated and was commissioned, or at some other time.
If Sarek stopped speaking to Spock when Spock graduated and was commissioned in Starfleet, that would leave only about five years for Spock to become the third officer of the Enterprise before "The Cage", and for the time between "The Menagerie" and "Journey to Babel", on either end of the thirteen years between "The Cage" and "The Menagerie". So making the eighteen years start any earlier in Spock's career is not recommended.
In another second season episode, "The Deadly Years", there is a competency hearing for Captain Kirk:
SPOCK: Medical banks, compute described subject's physical age, using established norms as comparative base.
COMPUTER: Working. Subject's physical age based on physiological profile, between sixty and seventy two. Aging rapidly.
KIRK: No, I'm thirty four. I'm thirty four years old.
So Kirk testified that he was thirty four years old, and he was not corrected even though his memory was going bad.
That means that Kirk was aged between 34.000 and 34.999 during that episode. If the five year mission lasted for between 4.000 and 5.999 years, Kirk's age range during various episodes must be less than from 28.001 to 40.998, and no more than 5.999 years between youngest and oldest age.
In "The Menagerie Part 1" Commodore Mendez said:
MENDEZ: You ever met Chris Pike?
KIRK: When he was promoted to Fleet Captain.
MENDEZ: About your age. Big, handsome man, vital, active.
If Mendez meant that Pike was about the same age in "The Menagerie" as Kirk was in "The Menagerie", that would make Pike aged about 28.001 to 40.998, which would make Pike aged about 14.002 to 27.998 in "The Cage" 13.000 to 13.999 years earlier.
Or maybe "about your age" means within 5.00 years of your age, which would make Pike aged about 9.002 to 32.998 in "The Cage".
Or maybe "about your age" means within 10.000 years of Kirk's age, making Pike about 4.002 to 37.998 in "The Cage".
That last age range reaches almost to Jeffrey Hunter's age of 38 when he acted in "The Cage".
So depending on how Mendez's statement is interpreted, Captain Christopher Pike might be aged about 14.002 to 27.998, or about 9.002 to 32.998, or about 4.002 to 37.998 in "The Cage", and if he was the same age as Jeffrey Hunter he would be 38 in "The cage".
Or Commodore Mendez's statement might be interpreted as meaning that he remembers Christopher Pike as being the same age when Mendez knew him years earlier as Kirk is in "The Menagerie". Since it is uncertain how long ago Mendez remembered Pike, Pike's age in "The Menagerie" and "The Cage" would be unknown.
If The Making of Star Trek, Stephen E. Whitfield, 1968, is canonical enough to be included in this discussion, it provides more information that is helpful. It gives the ages of some Star Trek characters and other chronological data. If all of that chronological information is accurate at the same moment of time, that moment of time can be narrowed down to within less than one year.
In "Who Mourns For Adonais?" in the second season:
CHEKOV: Perhaps if I assisted.
KIRK: How old are you?
CHEKOV: Twenty two, sir.
The Making of Star Trek, Stephen E. Whitfield, 1968, Part II: An Official Biography of the Ship and its Crew, chapter 8, Other Star Trek Regulars, says about Ensign Chekov: "All Ensigns seem universally twenty-two years old, but he is portrayed as being reliable and dependable, with a good head on his shoulders despite his youth." This can be interpreted as meaning that Chekov is still twenty two and that "Who Mourns for Adonais?" and maybe all second season episodes are less than one year before The Making of Star Trek.
In a scene in "Journey to Babel" Sarek says that his age is 102.437 of McCoy's years.
The Making of Star Trek, Stephen E. Whitfield, 1968, Part II: An Official Biography of the Ship and its Crew, Chapter 5, Mr. Spock, says:
Sarek is 102 years old, or about middle-aged, in terms of Vulcan years (The Vulcan life span is about 250 years)." Spock's biography also mentions events in "Journey to Babel" as past.
Therefore, the date of the chronological data in The Making of Star Trek should be sometime between about 102.437 to 102.999 YS (Year of Sarek).
Spock's biography also says that Amanda is 58. If Amanda is 58.000 to 58.999 sometime between 102.437 and 102.999 YS, Amanda should have been born about 43.438 YS and 44.999 YS.
Spock's biography also says that Spock and Sarek did not speak for 18 years before "Journey to Babel", agreeing with the episode, which implies that Spock probably graduated from Starfleet Academy and was commissioned in Starfleet about 18.000 to 18.999 years before "Journey to Babel" and thus about 83.438 YS to 84.437 YS, when Amanda should have been aged about 38.439 to 40.999.
At the present time the age requirements to enter US service academies are 17.000 to 22.999, meaning that cadets who graduate after the usual 3.75 years do so aged about 20.75 to 26.749. It is possible to graduate and be commissioned at younger or older ages than that. If Spock graduated and was commissioned aged about 20.75 to 26.749 in 83.438 YS to 84.437 YS, he would have been born about 56.689 YS to 63.687 YS, when Amanda was aged about 11.690 to 20.249.
Curiously, Spock's biography says that:
"Spock entered the Space Academy and, following his graduation eight years later, entered the space service."
Why would Spock take eight years to graduate from the Space Academy? I have to wonder if Whitfield wrote "6", "5", or "3" years in his notes and later misread that as "8" years. Or maybe Spock entered the Academy as a kid and graduated from the normal course after the usual 3.75 years or sooner, and then spent years more at the Academy taking postgraduate courses until he was old enough to be commissioned.
Spock's biography also says that:
"Spock has served aboard the USS Enterprise for 13 years, the first nine of them under Captain Christopher Pike and the last four under Captain James Kirk."
The Making of Star Trek, Stephen E. Whitfield, 1968, Part II: An Official Biography of the Ship and its Crew, Chapter 4, The Ship's Captain" says:
"Kirk has been in command of the Enterprise for more than four years and is the youngest Academy graduate ever to have been assigned as a Starship Command Captain."
I interpret those statements to mean that Kirk has been in command of the Enterprise for between 4.000 and 4.999 years at the time of The Making of Star Trek, and thus that Kirk took command of the Enterprise sometime between 97.438 YS and 98.999 YS.
In "Journey to Babel" Amanda says:
AMANDA: And you haven't come to see us in four years, either.
SPOCK: The situation between my father and myself has not changed.
If Amanda's "four years" is somewhere between 3.0 years and 4.999 years, the last time Spock visited Sarek and Amanda, presumably taking leave to do so, would have been sometime between about 97.438 YS and 99.437 YS.
In "Amok Time" Kirk says:
KIRK: I'm more interested in your request for shore leave. In all the years
SPOCK: You have my request, Captain. Will you grant it or not?
KIRK: In all the years that I've known you, you've never asked for a leave of any sort. In fact, you've refused them. Why now?
Kirk's years of knowing Spock might possibly have begun before Kirk took command of the Enterprise, but more likely began when Kirk took command. Since it would be hard for Spock to take leave to visit his parents after Kirk took command of the Enterprise without Kirk noticing that, Kirk probably took command of the Enterprise after the last time Spock visited Sarek and Amanda, but at least 2.000 years before "Amok Time".
Assuming that "Amok Time" happens sometime, but less than a full year, before Sarek is 102.437 in "Journey to Babel", "Amok" time should happen about 101.437 YS to 102.437 YS, and Kirk should take command of the Enterprise before 99.437 to 100.437 YS.
So the various date ranges when Kirk takes command of the Enterprise should be 97.438 YS to 98.999 YS, or 97.438 YS to 99.437 YS, or 97.438 YS to 100.437 YS.
If Pike commanded the Enterprise for nine years - 9.00 to 9.999 years - before Kirk took command, Pike should have taken command between 87.439 YS and 89.999 YS, or between 87.439 YS and 90.437 YS, or between 87.439 YS and 91.437 YS.
If Spock began serving under Pike about 11.347 years before Kirk took command of the Enterprise, that should have been between 86.091 YS and 87.652 YS, or between 86.091 YS and 88.090 YS, or between 86.091 YS and 89.090 YS.
Spock should have graduated from the Academy and been commissioned about 83.438 YS to 84.437 YS, as calculated above, and so that was a few years before he began serving under Pike.
If "The Menagerie" happens sometime between when Kirk took command of the Enterprise and when Sarek was aged 102.437 in "Journey to Babel", it should happen between 97.438 YS and 102.437 YS. If "The Cage" happens 13.000 years to 13.999 earlier, it should happen between about 83.439 YS and 89.437 YS.
If "The Deadly Years" happens less than a year before The Making of Star Trek, it should happen sometime between about 101.437 YS and 102.999, So Kirk should have been born about 66.438 YS to 68.999 YS.
So if Mendez meant that Pike was Kirk's age within a year, Pike would have been born about 65.438 to 69.999 YS.
If Mendez meant that Pike was within 5.00 years of Kirk's age, Pike would have been born about 61.438 YS to 73.999 YS.
If Mendez meant that Pike was within 10.000 years of Kirk's age, Pike would have been born about 56.438 YS to 78.999 YS.
If Mendez meant that pike had been about Kirk's present age an unspecified number of years earlier Mendez remembers him from, Pike's age would be uncertain.
If Pike was 38.000 to 38.999, the same age as Jeffrey Hunter, in"The Cage", happening between about 83.439 YS and 89.437 YS, Pike would have been born sometime between about 44.440 YS and 51.437 YS.
If Spock graduated and was commissioned aged about 20.75 to 26.749 in 83.438 YS to 84.437 YS, he would have been born about 56.689 YS to 63.687 YS, when Amanda was aged about 11.690 to 20.249.
So if Pike was born sometime between about 44.440 YS and 78.999 YS, and Spock was born between 56.687 YS and 63.687 YS, Pike would be somewhere between 19.247 years older than Spock and 22.312 years younger than Spock.
How do more recent Star Trek productions depict Christopher Pike and Spock?
Star Trek (2009) mostly happens in 2258 according to he official but not necessarily correct chronology, which would partially overlap with year 93 YS.
Christopher Pike was portrayed by Bruce Greenwood, born 12 August 1956, and Spock was portrayed by Zachary Quinto, born 2 June 1977, who were about 51 and 30 respectively during filming from 7 November 2007 to 27 March 2008. If they were those ages in year 93 YS they would have been born about 41/42 YS and 62/63 YS respectively, and Pike would be about 21 years older than Spock. Pike thus should have been about two years older in "The Cage" than Jeffrey Hunter.
In the Star Trek Discovery episode "If Memory Serves" 7 March 2019, Captain Christopher Pike is portrayed by Anson Mount, born February 25, 1973, and Spock by Ethan Peck, born March 2, 1986. So if Pike and Spock were exactly the same ages as their actors, Pike would be 13 years and 5 days older than Spock according to Star Trek Discovery.
Pike and Spock would be about 46 and 33 in "If Memory Serves", which I think probably has a date of about 2255 to 2256 in the official chronology, corresponding at least partially to 90 YS to 91 YS. So they should have been born about 43 YS to 45 YS and 56 YS to 58 YS respectively.
Memory Alpha says that Pike attended Starfleet academy sometime before 2220 in the official chronology, which would correspond roughly to 47 YS. But that makes Pike much older than other indications, and does not have a source indicated by Memory Alpha.
But Memory Alpha also says:
According to his biography on the Star Trek movie app, Pike enrolled in Starfleet in 2223 and was commissioned as an officer in 2227. He served aboard several vessels, including the USS Olympus (β), the USS Aldrin (β), and the USS Yorktown (β).
If Pike was commissioned in 2227 (corresponding to 62 YS) aged 20.75 to 26.749, he would have been born about 2200.25 to 2207.249 (corresponding to 35 YS to 42 YS, which barely overlaps with his birth range calculated by other means.
According to various calculations made from TOS date, Pike would be somewhere between 19.247 years older than Spock and 22.312 years younger than Spock, but probably older than Spock. Comparing the birth dates of various actors who portrayed Pike and Spock together, Pike should have been 5, 13, or 21 years older than Spock.
Added May 10 2019 And see also my post number 13 in this thread: https://www.trekbbs.com/threads/amok-time-a-true-classic.299658/11