2264, according to the official Star Trek web site and the Star Trek Encyclopedia
According to James T. Kirk's file on StarTrek.com (the official site of the franchise), he was promoted to captain and took command of the Enterprise in 2264. His first five-year mission ended in 2269. (Source)
The above is the final word on this, but it is worth noting that this is corroborated by the Star Trek Encyclopedia, compiled by Michael and Denise Okuda, long-time Star Trek production designers (especially during the TNG era):
- Google Books preview of ST Encyclopedia (linked to relevant section)
As Discovery is set in 2256, it would be another 8 years before Kirk takes command of the Enterprise.