In your question there are a few errors so I'd like to correct them first.
Firstly, Star Trek: The Original Series which featured Kirk and Spock, was sent on a five year mission, not a three year mission.
In 2265, the Enterprise was assigned to a five-year mission of
deep-space exploration, and command passed to James T. Kirk. The
ship's primary goal during this mission was to seek out and contact
alien life. Captain Kirk's standing orders also included the
investigation of all quasars and quasar-like phenomena.
Secondly, this was not the first five-year mission for the Enterprise, let alone The Federation. Prior to Kirk, the Enterprise had also been commanded by Captain Robert April and Captain Christopher Pike.
Thirdly, the Enterprise did not have what was considered to be 'new' warp technology; the Warp Drive was invented back in the 21st century with it's first warp flight on April 5 2063.
Now, to your actual question about Kirk knowing of Archer. It seems in the original timeline (i.e. The Original Series) there is no reference on either the Memory Beta or Alpha pages about Kirk and Archer being in contact with one another. Presumably, seeing as though Enterprise is set following TOS, Kirk was aware of Archer, but this is not confirmed.
However, in the alternate reality (2009 reboot) we know he does. Consider the following quote from Memory Alpha:
He [Kirk] quickly learned that Scott was wiser than he lead on, after first
impressions belied the fact that he had begun to postulate the theory
of transwarp beaming while still on active duty (misplacing Admiral
Archer's beagle Porthos in the process, earning his isolation to Delta
As for the prime reality though (the one from TOS) consider the following quote about Archer:
Considered by historian John Gill, as the "greatest explorer of the
22nd century," Archer is widely credited with saving Earth from the
Xindi and paving the way for the founding of the United Federation of
Planets. He went on to become one of the Federation's earliest
presidents. (ENT: "In a Mirror, Darkly, Part II")
Unless Kirk is completely ignorant of Federation, let alone Earth, history, he would know of Archer.