Short Answer, no definite statements. Long answer, a few possible hints or clues
In "Shore Leave", sometime after stardate 3025.8, Kirk meets Finnegan from when Kirk was at the Academy:
FINNEGAN: I never answer questions from plebes, Jimmy boy.
KIRK: I'm not a plebe. This is today, fifteen years later. What are you doing here?
So apparently Kirk was a plebe, in his first year at the Academy, about 15.0 to 16.0 years before "Shore Leave" and should have graduated and been commissioned an ensign about 12.0 to 13.0 years before "Shore Leave".
In "A private Little War", around stardate 4211.4, Kirk says he made a planetary survey as a lieutenant thirteen years earlier.
In "Obsession", stardate 3619.2, it is said that Kirk was a lieutenant eleven years earlier.
Presumably it took Kirk a number of years to rise from lieutenant to commander of the starship Enterprise.
Thus there is an earliest possible date indication relative to the second season.
There are also indications of the latest possible date Kirk could have become commander of the Enterprise.
TOS episodes may have happened in:
1) production order.
2) stardate order.
3) broadcast order.
Or possibly in some other order or no order at all.
The first episode to mention the five year mission in the introduction was:
1) in production order "The Corbomite Maneuver" beginning on stardate 1512.2 and in 2266 according to the official but not necessarily correct chronology.
2) in stardate order "Mudd's Women" beginning on stardate 1329.8 and in 2266 according to the official but not necessarily correct chronology.
3) in broadcast order "The Man Trap" beginning on stardate 1513.1 and in 2266 according to the official but not necessarily correct chronology.
The first episode where Kirk was in command of the Enterprise was the second pilot "Where No Man Has Gone Before" - which does not have an introduction mentioning the five year mission - beginning on stardate 1312.4 and in 2265 according to the official but not necessarily correct chronology.
In "Where No Man Has Gone Before" Mitchell makes a tombstone for Kirk inscribed
"James R. KIrk C 1277.1 to 1313.7". It has been speculated that 1277.1 was the stardate of Kirk's birth, thus making about one year per stardate unit in the stardate system used at the time. So Kirk would have commanded the Enterprise for thousands of years during TOS if that stardate system continued for the rest of the series.
It has also been speculated that 1277.1 was the stardate when Kirk took command of the Enterprise probably a relatively short time before "Where No Man Has Gone Before".
Someone who accepts the theory that 1277.1 was the stardate when Kirk took command of the Enterprise and also has a theory about the relationship between Earth time and stardates can calculate when they think that Kirk took command relative to other episodes.
If the TOS episodes happen in stardate order and the "five year mission" lasted between 5.0 and 6.0 years from "Mudd's Women" stardate 1329.8 to stardate 5943.9, the second stardate in "All Our Yesterdays", then there would be 769.016 to 922.82 stardates per year and 2.105 to 2.526 stardates per day.
If the TOS episodes happen in stardate order and the "five year mission" lasted between 4.5 and 5.5 years from "Mudd's Women" stardate 1329.8 to stardate 5943.9, the second stardate in "All Our Yesterdays", then there would be 838.927 to 1025.355 stardates per year and 2.296 to 2.807 stardates per day.
Using such stardate ranges it would be easy to calculate how long stardate 1277.1 was before any particular episode with a stardate.
What if someone considers TAS to be part of the five year mission? Then known stardates of the five year mission would range from 1254.4 (before 1277.1) in "The Magicks of Megas-Tu" to 7403.6 in 'Bem", 6149.2 stardates. In 4.5 to 6.0 years there would be 1024.86 to 1366.488 stardates per year and 2.805 to 3.741 stardates per day.
Of course if the five year mission began before the first episode's stardate and and ended after the last episode's stardate there might be many more stardates per year and per day.
In "Menagerie part 1", stardate 3012.4 and 3012.6, and "Menagerie Part 2", stardate 3013.1 and 3013.2 Spock says several times that the events on Talos IV were thirteen years earlier - and thus between 13.0 and 14.0 years. Spock also says that he served with Captain Pike for:
SPOCK: Eleven years, four months, five days.
Right after Kirk says:
KIRK: I took over the Enterprise from him. Spock served with him for several years.
If "Eleven years, four months, five days" equal approximately 11.3415 Earth years, and if "thirteen years" equal about 13.0 to 14.0 Earth years, then Kirk must have commanded the Enterprise for at least 1.6585 to 2.6585 years before "Menagerie". If Spock began serving with Pike during the mission to Rigel VIII and Talos IV. If Spock began serving with Pike before the events of "The Cage" then kirk would have commanded the Enterprise for a longer period before "Menagerie".
In "Who Mourns for Adonais?" stardate 3468.1, Kirk asks Chekov's age:
KIRK: How old are you?
CHEKOV: Twenty two, sir.
The Making of Star Trek Stephen E. Whitfield, 1968, Part II, Chapter eight, says: "All Ensigns seem universally twenty two years old,...", thus implying that Checkov is still 22 and The Making of Star Trek is less than one year after "Who Mourns for Adonais?"
In "Journey to Babel" between stardates 3842.4 and 3843.4, Sarek says his age is:
SAREK: One hundred two point four three seven precisely, Doctor, measured in your years. I had other concerns.
The Making of Star Trek Stephen E. Whitfield, 1968, Part II, Chapter five, says: "Sarek is 102 years old, or about middle-age in terms of Vulcan years (the Vulcan life span is about 250 years)."
If the same type of years is used both times, the chronological data in The Making of Star Trek should be valid less than 0.563 years after the scene in "Journey to Babel".
The Making of Star Trek Stephen E. Whitfield, 1968, Part II, Chapter five, also says: "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 of them under Captain James Kirk." Thus Spock should have begun serving under Kirk on the Enterprise when Sarek was about 98.437 to 99.
The Making of Star Trek Stephen E. Whitfield, 1968, Part II, Chapter four, also 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."
If Kirk became commander of the Enterprise about 4.0 to 5.0 years earlier, that would have been when Sarek was aged 97.437 to 99.
If Spock began serving with Pike on the Enterprise 13.0 to 14.0 years before The Making of Star Trek, and if the voyage to Talos IV in the "The Cage" was 13.0 to 14.0 years before "Menagerie", then "Menagerie" should be minus 1.0 to positive 1.0 years before The Making of Star Trek, and that seems too short a time span.
But if Spock served with Pike for 11.3415 years, and on the Enterprise with Pike for nine years, that leaves a period of about 2.3415 years when Spock would have served with Pike while not on the Enterprise. And if some or all of those 2.3415 years was spent sometime after the voyage to Talos IV in "The Cage", while the Enterprise was being refitted and repaired perhaps, then "Menagerie" might be between 2.3415 and 3.3415 years before The Making of Star Trek.
So people who consider The Making of Star Trek canon or semi canon have another clue to the possible date when Kirk took command of the Enterprise.
Short Answer, no definite statements. Long answer, a few possible hints or clues
