Ciao Julian,
Once you arrive in Livorno, you have several options beyond the excursions offered by the cruise company (which do tend to be ridiculously expensive!).
You could
1. rent a car and drive yourselves to Florence
2. rent a car with driver - driver at your beck and call
3. take the train into Florence - you do have to follow timetables to you're back on ship on time
OR
4. take a tour from Livorno but with local companies (not the ones offered by the cruise ship) which are cheaper - they also make sure you're back in time for your ship at the end of the day.
For this last one, here's a list which includes tours to other parts of Tuscany:
http://www.partner.viator.com/en/83...-in-Livorno-Day-Tours-of-Tuscany-Florence/ttd --- there are also private shuttles to Florence in a minivan (so similar to #2 above but shared with others, a cheaper option)
As far as your question - which pertains to #1 above - you cannot drive into the
center of Florence but you can get pretty close and park and then walk. I just think the bigger hassle of picking up the rental and driving, figuring out parking might be too much for a day trip. Many of the cruises only give you something like 8 hrs at the very most to do everything! Even the train option makes it a little harder, just because there are specific timetables to wait for (and ride takes longer than by car).
So if I was in your shoes, I'd go with a private shuttle if you just want to come in Florence and do your own thing OR a tour such as this one:
http://www.partner.viator.com/en/83...d-Florence-Private-Day-Trip/d519-5292LIVORNO1
Hope this helps!