Not according to the original series. In "Shore Leave", Kirk sketches himself as an Academy nerd, who had a brief affair with a (slightly older?) woman before he was forced to choose between their relationship and Starfleet.

Off the top of my head, there are references to his working his way up the ranks in "Obsession" and "Dagger of the Mind".

He's much, much too young to be left in command of the Enterprise.