There is an excellent dedicated airport train called the Leonardo Express that shuttles between Fiumicino Airport and Termini Station in central Rome. The trains run every 30 minutes.
The cheapest transfer between Fiumicino Airport and Rome city centre is by bus. Terravision operate a regular dedicated airport bus service but it is not the only provider of bus services to Rome.
A classic shared shuttle service between the airport and hotels in Rome in a van with multiple drops, providing a transfer solution priced somewhere between public transport and a taxi/private car.
Rome city taxi's are regulated with a stipulated fare between the airport and within the city walls. The alternative is a private car with English speaking driver who will meet you in the terminal.
Rome Fiumicino airport is near the coast, west of Rome city centre about 25 kms away. There is one hotel where you can walk to the passenger terminals and a few other hotels with shuttles available to take you to the airport terminals.
There are two bus lines competing head to head between Ciampino and Rome's Termini Station in the centre of Rome. Both are dedicated airport buses, not stopping local buses.
Rome city taxi's are regulated with a stipulated fare between the airport and within the city walls. The alternative is a private car with English speaking driver who will meet you in the terminal.
Very frequent trains run between Civitavechia and Rome's Termini Station offering the cheapest transfer for independently minded travellers.
A private car with English speaking driver offering a personalised private transfer between your cruise ship and Rome's hotels or airports is the stress free transfer option.
The classic shared van shuttle operating between cruise ships and Fiumicino Airport or Rome city centre hotels with optional sightseeing tours on the way.
Naples has a complex public transport that includes buses, metro, trams, funicular railways, normal railways and boats - thankfully with a unified ticketing system.
The Naples ferry and cruise ship terminal are in the heart of Naples at Mollo Beverello. The airport bus starts at the ferry terminal and services both Central Station and the airport.
Naples airport is quite close to the city itself, just north-east of the city centre. There are bus links that service Naples as well as the nearby regional towns like Sorrento.
Sorrento has quite good public tranport combining local buses and the Circumvesuviana train service to Pompeii and Naples as well as fast boats covering the Bay of Naples.