Axanar is a planet mentioned in TOS 3rd seasons episode "Whom Gods Destroy" :

Kirk to Garth Of Ixar: I agree there was a time when war was necessary, and you were our greatest warrior. I studied your victory at Axanar when I was a cadet. It's still required reading.[...] But my first visit to Axanar was as a new-fledged cadet on a peace mission. [...] They were humanitarians and statesmen, and they had a dream. - A dream that became a reality and spread throughout the stars,
a dream that made Mr. Spock and me brothers.

Kirk was honored later with the "Palm Leaf of Axanar Peace Mission" (see TOS: "Court Martial").

At first it is important to find out something about Kirks life:

James Tiberius Kirk was born 2231 or 2232 in Iowa (see StarTrek IV.)

A family friend named Mallory, whose son later served on the USS Enterprise as a security officer, helped gain Kirk entry to Starfleet Academy. (see TOS: The Apple)

Spock spent 3 years at StarFleet-Academy, but in general it is a four-year institution (see TNG: The Drumhead, DataLore, The First Duty).

Kirk became the first (and to this date only) cadet to ever beat the no-win scenario "Kobayashi Maru". After taking the test (and failing) twice, Kirk took the test a third time after surreptitiously reprogramming the simulation computer to make it possible to win the scenario. Kirk got a commendation for "original thinking". (see 2nd StarTrek movie).

Wesley Crusher was 17 years old when he first tries to enter the Academy - and I couldn't find any entry, that a human joined earlier. Therefore Kirk entered the Academy in 2248 or later and graduated 2251 or later.

Kirk's first assignment after graduation was aboard the USS Farragut (see TOS: "Obsession"). There he was promoted to Lieutenant.

McCoy to Kirk: I was speaking of Lieutenant James T. Kirk of the starship Farragut.
11 years ago, you were the young officer at the phaser station when something attacked.
According to the tapes, this young Lieutenant Kirk insisted upon blaming himself.

2nd season TOS: "Obsession"

This was said 2266 and happened therefore 2254 or 2255.

Later he served as an ensign on the USS Republic NCC 1371 along with his friend Ben Finney. During a duty shift, Finney accidentally left a circuit open in the main reactor; Kirk logged this major error, and Finney was denied promotion. (see TOS: Court Martial).

Kirk: So it seemed to brash young Lieutenant Kirk on his first planet survey. [...] When I left there 13 years ago...

(from TOS 2nd season: "A Private Little War")

The statement above was said by Kirk 2266. And the first planet survey above led by Kirk must have been sometimes between 2252 and 2254.

To sum that up: Kirk served first at the USS Farragut; was promoted to Lieutenant and was transfered 2254 toe the USS Republic.

Before that Kirk was already Lieutenant teaching at the Academy, where he met the young Garry Mitchell - that happened "15 years ago" (see TOS-pilote: "Where no man has gone before" - equal to the second half of year 2264). These 15 years could be read as "nearly 15 years": In that way Kirk must have been a teacher ("in his class, you either think or sink") sometimes between 2249 and 2251.... but... it was said before, that he graduated 2251 or later?

As we have seen, Picard was so impressed by young Wesley Crusher, that he granted a civilian a provisional field commission as Ensign.

Kirk was an Ensign after the Academy - but maybe he wanted to stay to get a second chance at the Kobayashi Maru-test. That is why he stays ther and helped the teachers and helds courses for beginners. To be more accepted, he is for a short time honored as a Lieutenant.

In that case he was born 2231, entered Academy 2248 and needed three years. In 2251 he visited Axanar as "a new-fledged cadet on a peace mission", get honored and graduated - but failed at the Kobayashi Maru-test. Kirk staid at the Academy teaching lessons. One of his scholars was Garry Mitchell, whom he first met one or two years before. After that he served on the USS Republic with Ben Finney and 2254, as a young Lieutenant Junior Grade, he made his first planet survey on Tyree's planet.

Kirks mission to Axanar therefore happened 2251. And before that, he studied as a cadet the victory at Axanar of Garth of Izar. Therefore the battle of Axanar was before that and there is a connection between the two missions, because they are on different opinions about the role of the people of Axanar.

The Actor of Fleet-Captain Garth, Steve Ihnat, was born on August 7th 1934; and the man behind Kirk, William Shatner, was born March 22nd 1931. Garth should be elder as Kirk - but he is not a human from Earth. He came from the planet Izar (this naming pattern was also used for Sarek of Vulcan). Maybe the people of Izar are not aging that fast - or Garth changed his appearence; a technique he learned from Antos IV, which inhabitants are masters of cellular metamorphosis and instructed him in such methods. (see TOS: "Whom Gods Destroy")

Kirk learned about Garths battle at Axanar as a young cadet and went later in 2251 to Axaner himself. Therefore the Federation won at this planet 2249 (2248 is too early; 2250 cannot be, because Kirk said "later" and not "one year later").