It helps if you post the names of the ports if you want faster answers, but I will look up the cruise for you...
1 Rome (Civitavecchia), Italy 8:00PM
You will likely fly into Rome, stay a few nights and see the city, take one of the double decker tour buses and plan on making it a full day.
2 Bonifacio (Corsica), France 8:00AM 6:00PM
This small island has one main city which you can explore on foot. It has delightful views and plenty of shopping, but like most of Italy the shops are closed from noon to 3:30. The restaurants are open, however. You will need to take a shuttle or taxi to get there = not expensive.
3 Portofino, Italy 8:00AM 6:00PM
Also a lovely small town you can explore on your own - no tour necessary.
The ship shou;d drop you off right in the main square.
4 La Spezia (Cinque Terre), Italy 8:00AM 6:00PM
Make arrangements (bus, tour or train) just to get to Cinque Terre and back. Once you are there you do not really need a guide. This national park has a paved hiking trail along the coast that extends about 10 miles.
I believe La Spezia has the train station that takes you to Cinque Terre.
Take the train to the furthest point you want to see - walk, and then get back on the train in the next town. There are (I believe) four train stops in towns within Cinque Terre.
5 Florence/Pisa (Livorno), Italy 8:00AM 6:00PM
Defintely go to Florence - whether by train or by tour. Or rent a car. I also think Pisa is worth seeing, though many call it a tourist trap these days.
6 Amalfi/Positano, Italy 12:00PM 10:00PM
I don't know this port - sorry. I assume a drive along the Amalfi coast is offered, That is worthwhile.
7 Sorrento/Capri, Italy 8:00AM 6:00PM
Go to the Isle of capri and see the blue-green waters. Expect large crowds.
8 Taormina (Sicily), Italy 8:00AM 4:00PM
You can just walk around - or take and island tour. I have not been here in 25 years so I will leave it to others to comment.
9 Corfu, Greece 10:00AM 6:00PM
You will dock in the capital city. You can walk to shopping or If you want to take an island tour you will see some beautiful greenery and nice beaches. Not many activities in this island, but beautiful scenerey.
10 Kotor, Montenegro 8:00AM 2:00PM
It is a short visit - I don't how close the ship docks to activity, but with a short stop like this it usually indicates there is a nearby attraction they will offer a tour to. Take it if it sound interesting.
10 Dubrovnik, Croatia 6:00PM
11 Dubrovnik, Croatia 1:00PM
Dubrovnik by night! This small, midieval city can be seen on foot. Have dinner in a local restaurant at night, then wake up early and get out to see the local market in full swing. No tour necessary, except getting to the city itself.
12 Venice, Italy 9:00AM
You can walk to all the sites in Venice - walk from the ship to the trainstation - walk through it to the main canal and get a day pass on the Vaporeto, or just take one boat to St Marks Square and follow the main route back to the train station. This will get you through all the good parts of Venice except for the Universita. You might want to go there first from St Marks square, then go back and walk back to the train station.
13 Venice, Italy -- I hope you plan to stay a few days in Venice. For convenience of getting your luggage to the airport you should consider staying near the train station. There are cartage companies that will pick up your luggage for free (IF YOU ARE CLOSE TO THE STATION) and drive you to the airport in a van. Cost is about Eu:80/couple, a water taxi will be Eu:120 and will take you longer to get to th airport.