cruise to italian riviera a lot of questions

Rick Steves has a Med Cruise guide book you might want to check out. Also, if you haven't joined, Cruise Critic is the sister to TripAdvisor but for all things cruising. It has 'boards' like forums, and one section is all about ports with questions and tips. You can also find port excursions for each port by google. The thing to remember is that ports can be changed, delayed arrival, or missed entirely. If you take a tour focused on cruise passengers, they tend to come with guaranteed return to port times, some don't charge if you don't arrive. They are often cheaper than a cruise or private excursion, but you have pretty high port expectations in Naples, so you may benefit from a private guide and trip.
Cruise Critic also has a 'roll call' section to meet some of your fellow passengers. Often there people are arranging private tours with more bodies to reduce the costs.
Yes, Cinque Terre is from La Spezia.

When you stop at Nice, you could just take the bus or train from Nice to Monaco. I would recommend taking it past Monaco to see the Italian neighborhood of Menton and then stopping at Monaco on the way back. To me, Monaco was just a quick stop. Another option if interested is the Nice Food Tour - had a wonderful time with excellent food, learning some history of Nice, too. In our Food tour group, there was a couple from a cruise ship participating.

If you select Sorrento & Capri with no time for Pompeii, there is the option to see Ostia Antica at Rome. There’s so much to do in Rome!

If you google “shore excursions” for each of the ports, some good ideas will come up for your consideration. Also “things to do” in each port.

It's possible to see the Cinque Terre from La Spezia, but Rick doesn't recommend it for cruisers. The villages are tiny and cruise excursion groups overwhelm them. Consider alternatives such as Porto Venere or taking the train to Pisa or Lucca.

For Naples, you need to choose just one destination--it's impossible to see all three in 10 hours. There are many private excursion companies that will meet you at the dock, take you to a destination, and guarantee that you will be back in time to board your ship--and they're a lot cheaper than your cruise company's option. Check Cruise Critic or Rick's Mediterranean Cruise Ports book for some ideas.

Look at www.toursbylocals.com and pick out a private guide to take you where you want to go. Visitor is just a middleman selling services that you could buy yourself at a lesser price.

Some good suggestions so far. Also have a look at CruiseCritic.com . They have a ports of call forum which will discuss options in your ports. Also, join the Roll Call for your specific ship's sailing. You may find that your fellow cruisers are discussing excursions, or even organizing group excursions.

Don't forget that Naples is a historic place as well. The ferry to Capri is at least an hour each way, and Capri is a huge, highly-pedestrianized (taxi money is no help, that is) destination. In that sense, it is a poor choice for a cruise day. We did it from a Sorrento cruise day, which is a shorter ferry. On another cruise, we had Capri as the port. Sorrento is pretty missable from the cruise-stop point of view, but Pompeii is a superb visit, and very easy on your own with no local guides required. It's wise to allow four hours inside Pompeii. You need to be careful that you download ferry and Circumvesuviana schedules that are likely to apply your your specific April dates, because they could be "off season." (Circumvesuviana does not require reservation, only referring to the danger of missing the cruise departure if "Southern Italian" errors and omission result in your missing the specific train you target for return to the ship. (Metro, bus, or cab required to the Naples port from the train station.)

One sensible combination would be Pompeii and a street walk in Naples, with the Archaeology Museum if you have time. But if Naples is warm and lively in April (I have no idea ... ), I'd put walking and churches, etc. ahead of the Archaeology Museum - except for Art History majors.

We were on a cruise that stopped in Naples. I wanted to see Pompeii and my wife wanted to see the Amalfi Coast. We compromised and did an 8 hour tour that offered both. Big mistake. Too rushed in both locations and it felt like we sprinted instead of stopping and enjoying everything. My advice is to pick one, then maybe if you have time before you go back to the ship you spend some time in Naples at the museum or just exploring the city.

When we were in Naples we walked by passengers getting off a cruise ship. Their guide announced loudly that no one should go into the city of Naples! Rather sad to miss it and jump on a bus or ferry to other places.
Portofino is tiny. Have a quick look around and get a taxi or walk into lovely Santa Margherita Ligure.
In the Nice area , the one place I would skip seeing
is Monaco. High rise apartments, yachts.
Choose Villa Ephrussi and Cap Ferrat, Nice itself is wonderful or Antibes.

Cruise Critic has been recommended in several posts above, and I agree you should use it. However, it can be very hard to navigate.

Here is the link to their Italian Ports Forum: https://boards.cruisecritic.com/forum/464-italy-ports/

For your French stops, here is there France Ports Forum: https://boards.cruisecritic.com/forum/462-france-ports/

And here are their Roll Calls. There is a thread for every single individual sailing. You can find others going on your sailing and join private excursions they have set up. This is the answer to your conundrum - you get a smaller group than the ship's tour (say, 10-12 people), at about the cost of the ship's huge group tour. https://boards.cruisecritic.com/forum/20-roll-calls/

mj,

If you want to see the Cinque Terre, travel from La Spezia would be much easier, as the nearest of the five towns is only about nine minutes from there by train. If you're planning to go there on your own or as a couple rather than with a ~50 person cruise group, I don't see a problem with that. I'm not sure it's necessary to "book a tour" as the five towns are easy to see on your own. However, one tip.... be sure to allow adequate time to get back to La Spezia and the ship. The local trains are usually somewhat on time, but I wouldn't take any chances.

You can have a look at Portofino while docked there, but there's really not a lot to see. The restaurants seem to be a bit overpriced, and IMO the food wasn't anything special. You could hike up the hill to Castello Brown, as there might be some kind of art exhibition or whatever during the time you're there (good views of the harbour and town from the Castle). There will likely be a few yachts in port so you can have a look and imagine the lives of the rich & famous.

During your time in Rome, you might consider a daytrip to Orvieto. It's a beautiful Umbrian hill town and only about an hour or a bit more each way from Rome. Again, be sure to allow lots of time to get back to the ship.

During your time in Naples, there's no way you'll be able to see Pompeii, Sorrento and Capri in the short time you have available. I'd suggest forgetting about Capri and focus on Pompeii and Sorrento. That will be quite enough to fill your day. Rather than rely on public transit to get around, you could hire a private driver as that would be a more efficient use of time and he'd be able to drop you off right at the Naples docks.

As I recall, you can take a bus for a very reasonable cost from Nice to Monaco / Monte Carlo. You could also have a look at excursions offered from the ship.

There's lots of good info in Rick's Mediterranean Cruise Ports guidebook.

I don't know exactly where your ship docks in Nice, but once you get to the train station in Nice, you can easily travel to other Riviera coastal spots, including Monte Carlo. You don't need to take a tour or a cruise excursion to see coastal towns. The trains are relatively inexpensive and frequent. The part of Eze you want to see is a hill town; that will require a bus or a taxi. I don't think bus service is frequent, so you must be careful to be sure you get back to Nice in time for your ship's departure. If Eze is a definite must-see for you, I would try to head there first.

Personally, if in La Spezia, I would not miss taking the train over to Manarola at least. It’s easily accessible as the second town and the marina area was my favorite location in Cinque Terre. The trail around the headland to the north is a must. The cafe up there is a great stop for lunch. I would also prioritize Pompeii if you can swing it and haven’t been. I agree with earlier comments about a driver and I would get a guide because it’s easy to miss some very interesting bits like the mosaic floor and the ahead of their time infrastructure. More importantly, it’s easy to get fatigued on large areas that are quite similar to each other while missing gems.