Ask the hotel to call a taxi for you.
Why do you need to go that early?
The trip from downtown Naples to the airport by taxi is 10-15 minutes, at the most. The airport is just 3 miles from the city center (or less).
Take offs at NAP are not allowed before 7am (for noise abatement reasons, due to the vicinity to the city), therefore the earliest flights will push out of the gate at 6:55am.
If you are at the airport by 5:30, you will be more than early. I don't even think the checkin counters open before 5:30am.
EDIT: Just checked the website of the NAP airport. The check in hall opens at 5:30am and closes at 10pm. If you go earlier than 5:30a you might have to wait outside until they open the doors and you will be at the mercy of the bad Neapolitan airport night scammers!! Wooohooo! (that was meant to scare you and make you sleep an hour longer)