Yes, Kirk must know about Archer, and it is NOT implied in The Original Series that Kirk is the first to explore deep space with warp technology.

It is never suggested in The Original Series that Kirk's Enterprise (or that the generation of explorers that includes Kirk and his crew) are the "first ones" to explore deep space. It is also never suggested that warp technology is a "new technology". In other words, nothing in TOS contradicts the existence of previous generations of explorers who have access to warp technology.

Indeed, the episode "Balance of Terror" reveals that United Earth had once fought a protracted war with the Romulan Star Empire decades before the time of TOS, leading to the establishment of a Neutral Zone between their respective domains. The fact that humans could efficiently reach the location that became this Neutral Zone implies that warp technology was available well in advance of TOS.

Moreover, in the TOS episode "Metamorphosis" we learn that Zefram Cochrane had invented the warp drive for humanity, and that he had disappeared in the early 22nd Century. He was discovered alive and well by the Enterprise in the mid-23rd Century. This further cements the timeline: the creation of the warp drive predates TOS by over a century.

Finally, whether people like it or not, Star Trek: Enterprise is canon. In that television series, we discover that Captain Jonathan Archer was directly involved in the establishment of the Federation itself. Archer went on to be the Chief of Staff of Starfleet and, by 2184, the President of the United Federation of Planets.

Therefore, it is virtually impossible for James T. Kirk to not have known a great deal about Jonathan Archer.