Back to all this talk of Robert April and him being former Captain of a starship called Enterprise.
The NX-01 did fly around the Expanse for a hundred years once... so it's not a question of the ship not being spaceworthy. Generations onboard once kept it going, even if it did look in a right old state. But there's no way that Enterprise would still be in Starfleet service by the 23rd Century. It would be in the Fleet Museum like visitors going to see a space shuttle today and stay there well-preserved until the 24th, so that Deanna Troi can get it mixed up with all the other historical ships. Unless it were another NX class, or a replica and not the original.
That would be a crazier reference than a joke about a beagle, to try and squeeze in with any credibility. April looks to be in an earlier Starfleet uniform and presumably had a fresh off the production line Enterprise to command. Even if he were in charge of a trainee cadet vessel, that would still require something newer than the NX-01. So no sticking to anything we've seen established before... except Kirk's Enterprise was more than a decade later getting into space.
It'll be interesting what, if any of this makes it into Star Trek Into Darkness. Conspiracy theories and a cover-up involving a previous Enterprise being kept out of the history books, would be interesting as the plot alone. I hope this isn't going to involve coming across a three-hundred year old Botany Bay. That would begin to read more like convenient fan-fiction than bringing the NX-01 into all of this!