https://search.ricksteves.com/?button=&date_range=2y&filter=Travel+Forum&query=Amalfi+coast&utf8=%E2%9C%93
Hopefully the above Search will yield some helpful info. Plan on the AC being very crowded and hot in June.
Three days is not enough for the AC/ Sorrento area, especially from Rome. It's a four hour drive Rome toSorrento. The infrastructure south of Naples has sadly been neglected, the roads are constantly congested, the public transportation somewhat chaotic and "gritty". A car would be your worst nightmare because of the traffic and lack of parking. The area is a victim of its instagram success.
You could take a train to Salerno, but again, if those three days include two travel days, it's too much work for one full day of exploring.
Take advantage of any available shore excursions when you dock in Naples. The convenience would be worth it.
This is the year of the Pope's Jubilee, so Rome will be extra crowded. Secure your lodging asap.
And- you need to be back in Rome the night before your flight home, due to the transportation difficulties.
Is there another area closer to Rome you might be interested instead? Does the cruise offer extensions post cruise? These options may be worth exploring .
Have a great trip.