Upvote:2
What would be the best way to get to Napoli from Rome during these early hours?
Wait. As you've figured out, this is not really possible, short of a shuttle that will set you back $1000
The bus takes 4h, while the train takes at most 3h, I'd strongly recommend the train for that journey.
You can either take a nap in the airport and grab a train from the airport to the central Termini station in the morning to get the 6AM Italo train that arrives at 7AM
Or, you can take a bus to Termini directly (see this answer) on arrival and maybe get accommodation for that 6AM train and be fully rested before taking it