Unfortunately, May 1st is actually one of those official holidays where the majority of everything is closed as far as museums and exhibits go. Many shops will be closed as well. Restaurants, however, do remain open... to feed all the people who have the day off, of course ;-). Poor restaurant workers!
All jokes aside, the Palazzo Strozzi in Florence is one of the few that for sure will remain OPEN on May 1st. I believe Palazzo Vecchio museum is also another that has remained open in the past for May 1st but I'll have to check for confirmation on that as the date approaches. They won't be free, I don't think. The "Domenica Metropolitana" which has been reinstated this year offers free visits only to local residents on the first Sunday of the month. The "free first Sundays" at the State museums was changed to only be applicable for the months from October through March, but all this stopped with lockdowns. I have not seen any news that it was restarted this year, and at this point, it would not be applicable again until October of this year.
Generally, May 1st is spent outdoors on walks and picnics since many things are closed. I would recommend planning on something similar, enjoying the monuments and, if eating out, to call ahead and make reservations at restaurants.