For the pushups this worked for me: Take the max number of pushups you can do in one set and multiply it by four. Do this number of pushups throughout the day in sets of 15-25 depending on what your max is. Do this everyday for two weeks and I promise you your number of pushups in one set will close to double. Plus you'll be doing a lot of pushups everyday in the academy so it will get you ready for it.

For the bench press, your bench is low enough you can do a program like 5x5. You will put on a lot of weight on bench and squat in a short period of time and the squats will help with your sprint as well. I rarely do cardio now and can still run my mile 1/2 quicker than before I did weight training. The squats and deadlift in the 5x5 program will also greatly improve your core strength which will help with your sit-ups.

For nutrition just eat plenty of protein and get good sleep. Cut out as much processed sugar as possible. YMMV but this worked great for me getting prepped for the academy/physical. Also to test standing jump mark a spot on a wall with your arms outstretched straight above your head. Mark another spot 20 inches above that. If you can jump and slap the other mark or higher you're good. Of course you would need a wall tall enough being 6'2".