So you only have Sunday and Monday for Rome, and are leaving Tuesday? If so, the Vatican Museums are closed on Sunday so Monday would be the only day you'll have to do that. It's also the busiest day as the National Museums are closed on Mondays so they will be mobbed - not that they aren't anyway during high season.
But yes, you could do the museums and St. Peter's…with one caveat. The queue for tickets to the museums is going to be very, very long. As one of the most visited attractions in Rome, the way around standing in line for hours is to book your tickets - or a tour - in advance. I would get on the Vatican website RIGHT NOW and see if there's anything still open for available time slots, and book tickets if there is:
http://mv.vatican.va/3_EN/pages/MV_Home.html
With a good umbrella and the will to make the best of a soggy day, there are also heaps of fabulous churches you could visit as well; get out your guidebook, do some reading, and plot some of those on your map.