During a routine geological survey on Surata IV, Captain Kirk is accidentally struck by a sharp thorn growing on a vine plant. The away team immediately beams back to the U.S.S. Enterprise, where Doctor McCoy finds out that the thorn has released a deadly virus into Kirk's body. Within a matter of hours, the virus will reach Kirk's brain, killing him.
To try to save Kirk's life, McCoy puts him into a machine that will artificially stimulate his brain neurons, keeping them active and resisting the virus. This causes Kirk to dream of his past adventures aboard the Enterprise.
Then, to save money, the rest of the film is just clips of the first film, with new footage added to show McCoy and crew worried about Kirk, and then, a few minutes at the end, when Kirk recovers.
Or, the second movie could start with the Starfleet brass waking up the morning after the end of Star Trek, complaining of the hangover Romulan Ale gives you.
Admiral Richard Barnett: "Damn, I was wasted yesterday celebrating the fact that Earth wasn't turned into a black hole. I dreamed we promoted a cadet straight to captain and gave him one of our newest ships. A ship, that, coincidentally, is one of the only ones functioning since the others we sent to Vulcan were destroyed. How wasted was I?"
Commodore Christoper Pike: "Uh, sir, that was no dream..."
Cut to Enterprise to see Kirk dancing a la Tom Cruise in Risky Business through the corridors of the ship.
Captain James Kirk: "Suckers!"