Captain Kirk has a phazer and the Enterprise, but there’s one flaw for him. Without the rest of the crew he can’t operate it. He needs someone to set up the phasers and he needs someone to get it to move. While he’s doing that Largo could get aboard and take him out. Largo also has superhuman beer endurance because he drank 100 cans before passing out and is a genius. He could outsmart Captain Kirk. Plus he has a gun. Captain Kirk drops to 0-1. Largo wins.

Update! I’m afraid that Kirk manages to win this round with his phasers. Captain Kirk wins.