Since there is no canon factual information about the history of the ship, there are only two possibilities here:
1. The Enterprise-A was a brand-new ship.
2. The Enterprise-A was an older ship that was renamed.
Let's analyze the first possibility. It's highly unlikely that the ship was built specifically for Kirk and to replace the previous Enterprise. The former ship had only been destroyed three months prior, and in that time Kirk and his crew were considered renegades. There would be no reason whatsoever for Starfleet to "honor" Kirk with a new Enterprise, even considering the Whale Probe incident. That would mean that it was just a coincidence that the new ship had just completed construction and was ready to be launched after Kirk was exonerated. Plus, this brand new ship would have gone from construction to decommission in only 7 years, with no explanation actually given for why it was being decommissioned.
So, the second possibility. We have a ship that presumably had been built at least after the original Enteprise's refit in 2270, which would mean that it could have been at least 20 years old by the time of its decommissioning. Ironically, even though Admiral Morrow's math is off, he stated that the original Enterprise was going to be decommissioned because it was "twenty years old." So we have proof in dialogue that the late 23rd century Starfleet considers ships two decades old to be worthy of decommission. We also see an older-style TMP bridge interior at the end of STIV, before it is replaced by a newer bridge, presumably after a refit between STIV and STV.
So here's my take: The Enterprise-A was originally another, older ship, which was simply renamed and re-registered after Kirk was exonerated and demoted to captain. Starfleet was probably going to build a new Enterprise (presumably an Excelsior class) after the TOS Enterprise was decommissioned (and that ship would have been the Enterprise-A), but changed circumstances dictated what we eventually saw happen.
As for the ultimate fate of the ship, I would assume based on my above interpretation that it was probably scrapped.