There is that single shot the two posters above discuss: Kirk in gold shirt that has Captain braid, backslapping somebody and a voice(over?) saying "Let's buckle up!". That's the last bit in the second trailer.
It should be noted that the sequence in the trailer shows Kirk and McCoy as simultaneous cadets - but later shows McCoy as a Lieutenant Commander. So it's not just Kirk's career that moves forward. We have every reason, then, to think that a significant amount of time passes between the "Kirk and McCoy are cadets" scenes and the "McCoy is a Lieutenant Commander" scenes, hence giving time for Kirk to climb the rank ladder, too.