Of course you can improve in the APFT by running, pushing and sitting, but are there any other exercises that may do a more efficient job?

You're training for a test. That's what it is really about.

So, you focus on the muscle groups... or INTENT behind the test.

In the case of running, it's an ENDURANCE test. Speed is a component, but in reality, the goal is to test cardiovascular endurance. So.... Walk. Walk FAR. You can jog far as well. start doing a 4~ mile walk at chow and your run time will get better because you are able to stay on your feet longer. 2 miles (12-15 min) is easy because you can do 1 hour no problem.

Sit ups is the same concept. Use PLANKS. Focus on CORE strength. You want to feel a burn, try doing a 2 min plank 5~ times a day (every other).

Push ups. Add Pull ups/Chin ups to your regime. Remember all muscles PULL (Flex). Push ups work against the natural flow of the body. They are NOT an efficient form of strength building. They focus on the opposite muscle group to bring you up. Pull ups / Chin ups, work with your body to build up the body, and will make it easier on you for when you do have to do pushups.

In the winter, I use cross-country skiing. If there is a better workout, I don't know what it is.

Something that build up your overall stamina. You can try swimming, biking, rowing and a mix of crossfit