We visited many of those ports in 2019. We did not use any cruise line excursions, preferring to go our own way. Not everyone is comfortable with that for many valid reasons, but in case you or anyone reading this is, here is what we did:
Santorini - We booked a private tour through
https://santorini-private-guide.com/ . It included a great tour of the island, a nice long stop in Oia, a fabulous lunch at a beach front restaurant in the middle of nowhere, and we ended the night watching the famous Santorini sunset from a winery while doing a wine tasting.
Athens - We booked a private tour through
https://www.taxigreece.com/tour/acropolis-athens-city-sites-plaka/ (recommended by Rick Steves). We saw all the highlights of Athens. Our guide was terrific. I highly recommend this company!
Mykonos - this was the one stop on our cruise where we didn't book anything. We were on a 14 day cruise with only one sea day (day one) and we felt this was a good "wing it" day. It was a great choice. We had a wonderful morning just wandering the old town which is lovely. We did some shopping and had a nice lunch overlooking the sea. It was a national holiday in Greece, so there was a huge festival going on. It began to get really crowded, so after taking a lot of it in, we headed back to the ship and napped on our balcony

.
Naples - We booked a private tour with this company
https://www.lovelyamalfitours.com/pompeii-amalfi-coast. We toured the Amalfi coast, Positano, Sorrento, and Pompeii. We did spring for the extra cost of an extra guide for Pompeii which I highly recommend. Italy and Greece have some very strict laws regarding guides vs drivers. While most every "driver" we had was really a great guide, they are not allowed to go into the sights with you. They did a great job preparing us and giving us good information, but sometimes you need someone with you. Pompeii is one of those places.
Livorno - most people head to Florence here and rightly so. For us, it was the last night of our cruise. We felt after 13 straight days of ports we needed a bit of a break from the culture overload. Someone in our Cruise Critic roll call group organized a wine tasting tour through Tuscany, with a short stop in Pisa to take the obligatory picture next to the tower. It was the perfect way to end our cruise and we still order wine from one of the places we toured. The company they used was
https://www.guidos-tours.com/.
Hope this helps!!