My two biggest problems are:
- The distance Kirk and Scotty supposedly transported from the planet (apparently hours after Spock set Kirk adrift, travelling at warp all the while)
- The sheer ridiculousness of a person with three years' experience being made captain of the flagship at the end of the film, with outcast Scotty, fellow cadet Bones and 17-year-old Chekov all in senior positions on said flagship.
1) Yeah, that first one is a little bit of a stretch. Unless, OldSpock, who is from the TNG era, programmed in some futuristic changes because he's from an era in which interplanetary beaming technology (which I believe, was a plot of a TNG episode once) is functional. Maybe not utilized, due to some sort of Federation equivalent FDA standars, but possible. Eh, that's still a stretch, but...
2) This is a stretch, but it's easier to justify. Kirk is a genius hotshot, as Pike stated, he's got a great pedigree, and he had just finished saving the lives of everyone in that room, and on the planet. Starfleet isn't necessarily the US Navy, so while we look at it as "jumping ranks", they may think it's more like taking a qualified individual and putting him in a position he's qualified for.