This is a long answer.
In short, don't trust the official dates. Calculate Spock's dates from the information given in TOS, which is canon. Spock shouldn't be much older than Kirk, going by the apparent age of his mother Amanda.
Part One (of Three): Don't Trust the Official Dates.
The dates in the official chronology from Star Trek Chronology: A History of the Future and other official sources are not canon. Only the dates given in various episodes and movies are canon, and they are only as correct and/or precise as their explicit data is.
So there are hundreds of dates with years mentioned or seen on screens, etc. in various productions. However, as far as I remember, almost every single one of them is merely a year, or sometimes a year and month, or a year, month, and day.
Describing a year as a year number with no additional information is inherently vague since it doesn't make explict which calendar era is being used.
As far as I remember I counted only six dates explicitly mentioned as AD or BC and thus Anno Domini dates.
Every other mention of a year leaves the calendar era unspecified and thus uncertain.
Of course it is natural to imagine that characters in a fictional francise produced in the USA always used the same calendar era when mentioning years, and that the calendar era they us ies Anno Domini.
And so I would assume, and did assume for a long time, that all the dates in Star Trek use the Anno Domini calendar era, until the evidence that dates in various Star Trek productions are given using several different caldnedar eras became too strong for me to ignore and I realized that when the calendar era is not specified in Star Trek it should be considered uncertain.
In TNG "Encounter at Farpoint":
RIKER: Then your rank of Lieutenant Commander is honorary?
DATA: No, sir. Starfleet class of '78. Honours in probability mechanics and exobiology.
LAFORGE: This once was rich farmland. I'd say something like twenty to thirty years ago.
DATA: I was discovered twenty six years ago.
LORE: Promises he later proved to be true. Which made you and me possible, brother. Our beloved father. Will I soon have a uniform like that, brother?
DATA: If you get one the way I did, Lore, it will mean four years at the Academy, another three as ensign, ten or twelve on varied space duty in the lieutenant grades.
Combining those three statements produces a date range for the first season of TNG.
But the last episode of that season "The Neutral Zone" had a converstation with Ralph Offenhouse, revived after being dead for centuries:
RALPH: What year is this?
DATA: By your calendar two thousand three hundred sixty four.
Data tells Ralph it is the year 2364 in Ralph's calendar, a year which is outside the range of possible years calculated from the previous data.
Since dates given in later seasons seemed to line up with the dates in Ralph's calendar, it seemed obvious to me that the calendar era had been changed to readopt Ralph's calendar era, perhaps due to the fame of the revived dead people.
And once I realized that I noticed the use of dates in TOS so inconsistent that they should have used different calendar eras.
So when a Star Trek character mentions a year without specifying the calendar era used, it is uncertain which of many Earth calendar eras is used.
Part Two: Spock's Chronology Up to the era of TOS From Canon Sources.
In "The Menagerie Part 1":
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.
Eleven years, four months, and five days are approximately equal to 11.347 (eleven point three four seven) years.
Later Spocks shows the mission to Talos IV at his court martial:
SPOCK: This is thirteen years ago. The Enterprise and its commander, Captain Christopher Pike.
After interruption, spock resumes:
SPOCK: As I stated, gentlemen, this was thirteen years ago. We were on routine patrol when the ship's sensors detected something ahead. At first we were not certain what it was.
Kirk's narration at the beginning of "The Menagerie Part 2" says:
...How Spock could do this he refused to explain, but there before our eyes, actual images from thirteen years ago of Captain Pike as he was when he commanded this vessel, of Spock in those days, and of how the Enterprise had become the first and only starship to visit Talos Four...
So the events at Talos IV were thirteen years before "The Menagerie". Assuming that thirteen years is between thirteen and fourteen years we can calculate how long ago Kirk took command of the Enteprrise. If Spock began serving with PIke at the same times as the mission to Talos IV, Pike would have stopped commanding the Enterprise sometime between about 1.653 (one point six five three) and 2.653 (two point six five three) years before "The Menagerie". If Spock started serving with Pike sometime before the mission to Talos IV, Pike would have stopped commanding the Enterprise a corresponding amount of time before the calculated time.
Addition on Dec. 19, 2020
In "Amoke Time":
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?
"All the years" that Kirk knew Spock well enough to keep track of Spock's comings and goings were probably at least 2.0 (two point zero) years. Kirk said he met Pike once, when Kirk took command of the Enterprise. Spock served with Pike for eleven years before that, so it would be hard for Kirk to observe closely without meeting Pike. Since Spock served with Pike for 11.347 (eleven point three four seven) years before Kirk took command of the Enterprise, Spock should have begun serving with Pike at least 13.347 (thirteen point three four seven) years before "Amoke Time".
In "Journey to Babel" Amanda said to Spock:
AMANDA: And you haven't come to see us in four years, either.
Being human, Amanda might say four years when the actual time span was three years or five years. Since Spock probably had to take leave to visit his parents, the earliest that Kirk could have taken command of the Enterprise would have been soon after Spock returned from that leave.
If Spock visited his parents less than 5.0 (five point zero) years before "Journey to Babel", Kirk should have taken command of the Enteprise less than 5.0 (five point zero) years before "Journey to Babel". And therefore the 11.347 (eleven point three four seven) years that Spock served with Pike must have begun less than 16.347 (sixteen point three four seven) years before "Journey to Babel".
Since the voyage to talos IV must have happened after than spock began serving with Pike, it must have happened less than 16.347 (sixteen point three four seven) years before "Journey to Babel".
If the voyage to Talos IV happened immediately after Spock began serving under Pike, "The Menagerie" 13.0 (thirteen point zero) to 14.0 (fourteen point zero) years later could be no more than 2.347 to 3.347 years before "journey to Babel". And it could be less if Kirk took command of the Enteprise less than five years before "Jouney to Babel". And if the voyage to Talos IV happened sometime after Spock began serving with Pike, the possible time betweenn "The Menagerie" and "Journey to Babel" would shirnk by the same amount of time.
It is common to supposed that "Amok Time" and "Journey to Babel" happen within a few months time, although there is no absolute proof that is the case. Supposing that "Amonk time" and "journey to Babel" happened at the exact same time, then Kirk would have taken command of the Enterprsie sometime between 2.0 (two point zero) and 5.0 years before the time of those two episodes.
Since Spock began serving under Pike 11.347 (eleven point theree four seven) years before that, Spock began serving with Pike sometime between 13.347 (thirteen point three four seven) and 16.347 (sixteen point three four seven) years before those two episodes. If the voyage to tTo alsos Iv was immediately after Spock began serving with PIke, it would have been sometime between 13.347 (thirteen point three four seven) and 16.347 (sixteen point three four seven) years before those two episodes.
And so the Menagerie" would be sometime between -0.653 (minus point six five three) years and 3.347 (three point three four seven) years before "Amok Time" and "Journey to Babel". And if time passed between Spock beginning to serve under Pike and the voyage to Talos IV, "The Menagerie" would be before those two episodes by a corresponding less time.
Thus it is mathematically possible to have a chronology based only on canon data in which "The Menagerie" happens after "Amok Time" and "Journey to Babel".
End of Dec. 19, 2020 addition.
I note that in "This side of Paradise":
ELIAS: You've known the Vulcanian?
LEILA: On Earth, six years ago.
Possibly the Enterprise was being refitted or repaired at Earth during Pike's command and Pike and Spock were thus on Earth six years before the epeisode.
In "Journey to Babel":
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.
The eighteen years that Sarek and Spock did not speak togethr as father and son would have begun sometime during Spock's joining Starfleet, sometime between Spock first expressing a desire to join Starfleet and Spock being commissioned an ensign in Starfleet.
In: "The Enterprise Incident":
COMMANDER: How long have you been a Starfleet officer, Spock?
SPOCK: Eighteen years.
I know in the US services an officer's time in service was sometimes included his years at a service academy and at other times with other rules began when they were first commissioned after graduation. So this indicates that Spock either entered Starfleet Academy 18 years before "The Enterprise Incident" and graduated and was commissioned a few years later, or else graduated and was commissioned 18 years earlier and studied at Starfleet Academy for some years before that.
If the 18 years in "Journey to Babel"and "The Enterprise Incident" refer to the same event, the two episodes would be less than one year apart. But if Amanda referred to Spock entering Starfleet Academy and Spock referred to being commissioned., "The Enterprise Incident" could be about four years after "Journey to Babel".
In "Journey to Babel":
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.
So Sarek is 102.437 (one zero two point four three seven) years old at that moment of time. Thus Spock could easily be sixty years old in "Journey to Babel", except that his human mother, Amanda, doesn't look more than sixty herself. Unless Amanda has spent a lot of time in suspended animation, or time warps, or something, after Spock was born, and thus is a lot younger than the number of years which have elapsed since she was born, it is unlikely that Spock is as old as the forty something Scott or McCoy.
Part Three: More Spock Chronology, Using Information From TAS Which Might Not be Canon.
TAS is not considered canon by everyone.
In "Yesteryear" Kirk and Spock return from a journey to the past to find that Spock has been erased from history.
KIRK: I don't know what's going on, but the first officer of this ship will be treated with respect.
(An Andorian enters)
THELIN: Captain, I assure you no one has ever treated me otherwise.
KIRK: Who are you?
MCCOY: Oh, I thought sure you'd know Thelin by now, Jim. He's been your first officer for five years.
If Thelin became first officer to Kirk at the same time as Spock did in the "prime" universe, Spock should have done so about five years, more or less, before "Ysteryear".
Kirk learns that while He and Spock were in the past, someone else was using the Guardian of Forever:
KIRK: If we didn't change anything while we were in the time vortex, someone else must have. Was the Guardian in use while we were gone?
GREY: Yes, but it was nothing unusual. We were scanning recent Vulcan history.
SPOCK: What time period?
GREY: Twenty to thirty Vulcan years past.
So the change in the timeline happened twenty to thirty Vulcan years in the past.
And in another scene:
BATES [on monitor]: Yes, sir. I can relay that to your screen. (showing relevant images) Sarek of Vulcan. Ambassador to seventeen Federation planets in the past thirty years.
SPOCK: That is not correct.
So the change in history should have been thirty or more years earlier.
If the change was both twenty to thirty Vulcan years earlier, and at least thirty years earlier, it should have been thirty years earlier. If years of the same type were used every time.
SPOCK: My mother. The son, what was his name and age when he died?
BATES [on monitor]: Spock. Age seven.
And Spock tells the Guardian of Time"
SPOCK: I wish to visit the planet Vulcan, thirty years past, the month of Tasmeen. Location, near the city of ShirKahr.
So this establishes that Spock was probably aged about thirty seven in "Yesteryear" and probably about thirty two when he became Kirk's first officer about five years earlier.
Of course there is the problem that some of or all of the data is given in Vulcan years, which may be noticably longer or shorter than Earth years. In fact, Vulcan is usually believed to be a planet in the 40 Eridani system. There are three stars in the 40 Eridani system, and all are dimmer than the Sun. Planets in the habitable zone of even 40 Eridania A, the brightest, would have to be closer to their star than Earth is to the Sun, and thus have shorter years.
And some people do not accept TAS as canon.
Part Four: Using Data From The Making of Star Trek, Which Might Not be Canon.
The Making of Star Trek has biographies of the main characters and their actors.
The chapter on Spock says his father Sarek is 102 years old. It also says that his human mother, Amanda, is 58 years old. Amanda's age agrees with her age in the script of "Journey to Babel" and the age at the time of Jane Wyatt, although none of those is canonical data.
The Making of Star Trek mentions events in "Journey to Babel" as a past experience. Since it says Sarek is still 102 years old, less than 0.563 (zero point five six three) years must have passed between "Journey to Babel" and the time period of The Making of Star Trek. The Making of Star Trek says that Spock graduated from Starfleet Academy eight years after entering it, which seems a very long time. Possibly someone wrote a number 6, 5, or 3, and someone later misread it as 8.
Considerng that Spock was already the third in command of the Enterprise at Talos IV thirteen years before "The Menagerie", I find it hard to believe that Spock spent eight years at the Academy, and took several years to become third in command of the Enterprise, thirteen years before "The Menagerie", thus having entered the Academy more than twenty one years before "The Menagerie". Since Amanda could have been as young as fifty eight in "Journey to Babel" that would make her thirty seven or younger when Spock entered the Academy.
So if Spock was at least seventeen when he entered the Academy, Amanda would have been twenty or maybe years younger when Spock was born. I can imagine that if Earth has tabloid newspapters in that era they might have headlines like "Alien Ambassador Elopes with Teenage Earth Girl".
I prefer to believe that Spock was the precocious one, entering the Academy as a child, or that there is a typographical error in the number of years Spock spent at the Academy. Or possibly Amanda is 58 years old but was born more than 58 years ago, having spent years in a time warp or suspended animation since Spock was born.
The Making of Star Trek says that:
Mr. spock has served aboard the U.S.S. Enterprise for 13 years, the first nine of them under Captain Christopher PIke, the last four under Captain Kirk.
The chapter on Captain Kirk says:
Kirk has been in command of the Enterprise for more than four years, and was the youngest Academy graduate ever to have been assigned as a Starship Command Captain.
Those two statements indicate that Kirk becamed commander of the Enterprise sometime between four and five years before the moment that the chronological data in The Making of Star Trek come from, and that Pike was captain for nine years before that, his command beginning thirteen to fourteen years before that chonological moment, and Spock probably began serving with Pike about two and a third years earlier or about fifteen and a third to sixteen and a third years earlier than the time when Amanda is described as being 58.
If PIke was in command of the Enterprise at Talos IV, and if Pike commanded the Enterprise for nine years, and if the voyage to Talos IV was thirteen years before "The Menagerie", Kirk would have become commander of the Enterprise about four years before "The Menagierie", which thus would have been less than about one year before the point in time where the chronological data in The Making of Star Trek are valid.
On the other hand, if Spock began serving under Pike right before the voyage to Talos IV, Kirk would have become the commander of the Enterprise between about 1.653 (one point six fiv e three) and 2.653 (two point six five three) years before "The Menagerie". That would make "The Menagerie" about 1.347 (one point three four seven) to 3.347 (three point three four seven) years before the point in time where the chronological data in The Making of Star Trek are valid.
I have to admitt that I haven't updated my Spock chronology with any data from TOS movies, the era of TNG, the Kelvin verse, DISC, or the upcoming Strange New Worlds.