I can tell you two places that won't work (closed per their websites): Ostia Antica and the Pantheon. This is just a guess, but I think you'll find churches open for at least part of the day. I hope one of our more experienced Italian hands can comment on that.
It would be a good day for a walking tour (organized, if available, or on your own).
I enjoyed a visit to the Non-Catholic cemetery in Testaccio. According to its website, it will be open from 9 AM to 1 PM on the holiday, with last entry at 12:30 PM.
Another possibility would be a day-trip to Orvieto, but I would expect it to be relatively crowded with visiting Italians that day, and I haven't checked to see which, if any, of the actual go-inside sights will be open. There are a couple of good museums, plus of course the Cathedral.
Generally speaking, it has been my experience that the more commercial sites (which are not usually my favorites) are more likely to be open on major holidays than government-run institutions. You might check out Trip Advisor's Things To Do list to if there's anything that sounds intriguing. Most of the time TA provides a link to the official website, where you'll be able to check on the days and hours of operation.