I was there last year, I think , on a different cruise line. I got off the ship kind of earlyish (maybe 10 or so) and took the bus to Livorno (I think it was a free shuttle from the ship). Most things around the square were closed IIRC but I did see a couple shops opening and I stopped for gelato and coffee. Aside from that, and a bunch of tourists, I don't remember seeing much of anything open at all (a random church here and there).
You might be better going to Florence. I didn't care as I'd only planned a quick look around and a snack, which is what I got. Hope that helps. (I should have gone to Florence but I got a case of I'm too lazy to do much-itis. LOL)