The best way to improve on pushups is to do pushups.
The best way to improve situps is to do situps (and other ab strengthening exercises.)
The best way to improve your run time is to run.
I'd advise checking out the scores on the APFT and practicing it as mentioned above. The academy expects people to achieve around a 275 or higher. Anything lower will typically put you into a morning workout regimen that starts at 5:30 each morning. New Cadets have a bit of a reprieve coming in since they will be active the entire summer and then test again (for record) on one of their last days of Beast. I'd really just do the above exercises. When I in BCT at Sand Hill, the Brown Rounds encouraged us to do elevated pyramid pushups. (Start at 1 pushup with your feet on an elevated surface, maybe a chair or bed, and then do 2, then 3, 4, 5...all the way to 10. Then Do 10, 9, 8, 7...until you get to 1. This really increased my pushups within a short time period (2 weeks or so.) I was always told in the Army that abs can be worked out everyday. So do that - do as many situps as you can in two minutes for a baseline. Once you have that number, try hitting 30% of your max in 30 seconds. Do that 4 times and then do your 2 minute max as the 5th set. To improve running times - you should run. That's sort of self-explanatory.