If you want to visit the museum in Naples, I think you have time. Just start early.
You can take the Circumvesuviana train to Naples, leave your luggage at the luggage storage at the station (DEPOSITO BAGAGLI), then go to the museum. After a visit at the museum, go back to the station, retrieve your luggage, take a high speed Freccia train to Roma Termini (70 min.) followed by the Leonardo Express train from Roma Termini station to the airport (32 minutes). You should be at the airport at least 75 min. prior to departure (better if 90, since there are lots of business travelers at that time). To find out your train schedule options departing from Naples and arriving at FCO by 5:15pm (17:15), use the Trenitalia.com website.
Enter
From; NAPOLI CENTRALE
To: FIUMICINO AEROPORTO
Hour: enter any time (for example 11) to see departures after 11am.
Note that the train to the airport is a Regionale, therefore schedules are available for view now only for the 7 days going forward. Just use a date in the next week, they won't change when you travel.
If you don't want to visit the museum, then Frank's suggestion on the bus or the train option above are the best. There is also the possibility of hiring a private car with driver, but the cost will be at least 350€. It may not be a bad deal for 4 people though.