People, people.. it's the same Archer. Enterprise featured a temporal cold war. Star Trek XI featured a Romulan changing history with time travel. It is clear that this reference to Archer in the movie speaks of an untold story, that of Archer ending up some years or decades into the future, unaged. Hence he is alive during Kirk's era because he is not as old as he should be. It could be as simple as what happened to Scotty in TNG or it could involve all kinds of nefarious plots.. and we would know all about it if ENT had not been cancelled!!!!
I think we already have a plausible answer as to why Archer could live 150+
In
Divergence, Archer was infected with the Klingon augment virus in order to produce enough anti-bodies for a cure. Even Phlox at the time stated "I would have to expose you to the plague; I cannot predict what it might do to you." Even Klingons suffered the residual effect of being genetically ridgeless a century later. It is plausible even after being cured, Archers exposure to the augment virus allowed him to live well beyond a normal human's lifespan.
It was, after all, just an
augment virus.