Generally, the day before a start should consist of a light workout, maybe working on picks, taking some ground balls, and normal throwing with change up catch and some shortwork. In my experience at the college level, most guys will want to pen two days before a start although a few guys have liked to throw a light pen the day before a start. The main thing is to have a routine and try to stick with it.
As for fungoes, it will be a little different at the high school level, but there should still be enough guys to hit fungoes, that the starter should not need to hit them on his start day. He should stay in the dugout, and focus on his job at hand, that is pitching. The day before and after though should be no problem.