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.

([Source][1])

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][2].

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][3] or [Alpha][4] 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][5]:

> 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
> Vega).


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")

<h2>Unless Kirk is completely ignorant of Federation, let alone Earth, history, he would know of Archer.</h2>

  [1]: http://en.memory-alpha.wikia.com/wiki/USS_Enterprise_%28NCC-1701%29#Kirk.27s_five-year_mission
  [2]: http://en.memory-alpha.wikia.com/wiki/Zefram_Cochrane
  [3]: http://memory-beta.wikia.com/wiki/James_T._Kirk
  [4]: http://en.memory-alpha.wikia.com/wiki/James_T._Kirk
  [5]: http://en.memory-alpha.wikia.com/wiki/James_T._Kirk_%28alternate_reality%29