In short order? With what planes and what munitions? Outside a few airfields in Romania, Bulgaria and Greece, which absolutely cannot service the number of planes that would be required for such an operation, NATO would be forced to rely on long range strike missions. Assuming Turkey stays neutral, which frankly given the situation with Erdogan is very likely, you wouldn't be able to transit ships or fly planes through their air space.
Sure if given enough time and resources NATO could mass a sufficiently strong enough force to attack the Black Sea Fleet, but then we'd run into the problem that while the Black Sea Fleet only has a few combatants with sufficiently strong enough air defenses, that could be as you said sunk in a matter of hours with enough fighter craft, they absolutely have a very large, well equipped and well trained air defense force in Crimea.
Over the last half decade Russia has turned into Crimea into the same kind of military fortress that Kaliningrad is with layers upon layers of air defenses and anti-ship missile launchers. We're talking divisions of troops here armed with S-400s, S-300s, TORs and BUKs and long range cruise missiles, let alone fighter craft and drones.
https://euro-sd.com/2020/03/allgemei...ed-the-crimea/
That was in 2019, not what they have now or what they mobilized recently.
So yeah good luck with sinking their fleet when it's got all of that to help protect it and they would absolutely retaliate with their own long range weapons. Any war between Russia and NATO is going to come down to long range missiles, radar, jamming and so on and it will be quite a long and bloody affair and there are quite a few areas of their military where Russia not only has parity with the US in many key areas, it has parity with all of NATO combined,
especially in munitions: Seriously take the US out of NATO and it's a paper tiger despite the hundreds of billions spent. Europe has done a shit job on defense spending, not only in overall numbers but how they've spent their cash except for France ( oh the irony ).