Archer's Enterprise would be 82 years old as of 2233. I don't know offhand the design life of the NX class, but I'd be real surprised if it was over 40 years.
Granted, that would be pushing it a bit, but consider that in the Trek novels, the U.S.S. Lovell, a Daedalus-class ship, was in service for over a hundred years. It's at least feasible that the NX-01 could be the ship April is referring to.