Everything is open on Sundays - in Siena, you should visit the Palazzo Pubblico (the frescoes of Good and Bad Government are AMAZING - consider when they were made!) and them climb up the tower to look down into Piazza del Campo! Then head to the Duomo and buy the OPA Si pass so that you can visit the inside of the Church, the Baptistery, the museum (which has another view from above of Siena) and the crypt between the cathedral and baptistery. All of these are open 7 days a week year round, with only a few exceptions around Palio time, but that's in July and August so not a problem for you in May!
Enjoy Siena, I truly love it and am glad when people spend more than a few hours there -- if you have the chance, spend the entire day and soak in the atmosphere at dinner when any crowds are gone ;-)