Ciao George! Thanks for your question.
The Naples airport isn't connected by train. The Frecciarossa high-speed train from Roma Termini (the main train station in Rome) takes 1 hour and 12 minutes to reach Napoli Centrale - Piazza Garibaldi (the main station in Naples). From the Naples train station, you can take the Alibus shuttle to the Naples airport42. The shuttle bus runs every 15-20 minutes and takes 15 minutes to get to the airport.
So to start, you'd need to get transportation from your hotel in the Vatican to Termini station, which you could do via public transport (bus 64169 is a common one for this route - estimate about an hour due to wait times and any traffic you may encounter; even though the route is only about 5 km, Rome public transport is notoriously slow); taxi or Uber/private driver. You could ask your hotel where the closest bus stop is for bus 64, or if they can help you book a taxi. There's also an app called Free Now177 that you can use to book a taxi in Rome. If you have Uber on your phone you can book there.
Since the train ride is just over an hour, if you want to get to the Naples airport around 4 pm, you'd want to get to the Naples train station around 3 pm, just to be on the safe side. So get a train around 2 pm, which means you'd probably want to leave for Termini at 1 pm if you're going by public bus, or 1:30 pm if you're going by taxi or private car. That would give you about a half hour or so to navigate the station and find your train. If you feel less confident navigating the station, add extra time.
While you can't purchase tickets more than six months in advance, I've copied a screenshot for you of an upcoming Saturday schedule for Rome-Naples from our Italian train tickets site ItaliaRail50.
You can see there is a high-speed direct train at 1:53 pm that arrives in Naples at 3:03 pm or another leaving at 2 pm and arriving in Naples at 3:12 pm, both at a cost of $44 USD. The Alibus schedule40 shows buses leaving Naples Stazione Garibaldi (central station) at 3 pm, 3:15 pm, and so on every 15 minutes. You could get the bus at 3:30 pm or 3:45 pm and be at the airport at 3:45 pm or 4 pm.
You can purchase the Alibus shuttle ticket49 one-way for 5 euros online or you can buy it from an authorized seller in the train station or on board the bus.
Let us know if you have any other questions! We're here to help!
