Yeah, I think the given year is 2188. But using relative dating (i.e. Jim Kirk's age), it adjusts to 2244 - an almost perfect fit for the conjectural TOS Enterprise launch date of 2245.
I think the problem vs. the original movie opening would have been, the Enterprise in service earlier than that adjusted date, in 2233.
The problems vs. Enterprise and the rest of post-1988 Trek are numerous. Technology, the way the pre-TOS Federation is described, and even the idea that April, as much as I liked the character, would ever become captain (he was such a pacifict he turned command of the ship over to George during battle because he couldn't bear to fire at a living being)