I remember a line in TOS where Captain Kirk indicated he was 34 years old. That might have been a book, I can't be sure. Anyway, in Star Trek 2009, I remember Bones making fun that he was 17 years old - which would make him 20 in Star Trek Beyond, I guess. Do we have any clue to his age in TOS?
In the episode "Who Mourns for Adonais?" from season 2 of TOS, his age is established as 22, see the transcript here:
KIRK: How old are you?
CHEKOV: Twenty two, sir.
In the chronology established in the TNG era, this episode took place in 2267, so in the Prime timeline Chekov would have been born in 2245. But in the Kelvin timeline, the main action of the 2009 movie takes place in 2258, so if Chekov is 17 there, in the Kelvin timeline he must have been born in 2241. Both dates are after the date Nero arrived in the Kelvin timeline though (2233), so this can be chalked up to a change in the timeline.
Incidentally, you're correct in your memory of it being mentioned that Kirk was 34 at one point in TOS--this was established in another episode from the second season, "The Deadly Years", where Kirk's biological aging had been accelerated and there was this exchange:
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.
Since this episode also took place in 2267, Kirk was born in 2233.
