Naples to Rome airport for 6 AM flight

Have you checked to see what time the trains start running?
This bus runs from Naples to both Rome airports:
http://www.fiumicinoexpress.com/orari-partenze/?lang=en

I would go to Rome for the last night.

I agree. On any bus or even on a fast train, Naples is a long way from Rome and your flight out, and there are a lot of things that could delay your getting to the airport.

Is your flight from Rome the beginning of a long international flight? If so, you would be taking a large risk by spending the night before you depart so far from the airport you're departing from. If something causes you to miss that flight, you could easily have to buy a new ticket on the spot - which could be very expensive. This is less of a concern if you're just taking a short regional hop to somewhere else, but do factor into your planes the consequences of missing your flight from Rome.

Personally, I would get to Rome the night before your departure and sleep there.

It looks like we could get a bus from Naples to Rome airport at any hour of the morning on this site http://www.fiumicinoexpress.com/faq/?lang=en

I do not believe you will catch your 6:20 plane if you rely on a bus from Naples on the morning of the flight. Take a train to Rome the night before and stay near the airport. Of course, like Dirty Harry asks, "Do you feel lucky? Well, do you?"

The only way you can make the flight is if you take the 1:45 a.m. bus. I've looked and can't find details (other than opinions) on how far beforehand you need to arrive to get through all the pre-flight stuff.
Your decision whether it is worth it all.

I posted on your other thread. You need to take the train the night before if you are going to do this:
https://community.ricksteves.com/travel-forum/italy/italy-to-greek-island-flight-rescheduled

Bad idea, sleep in Rome. If it’s business, let them pay for the change. Or pay the late night surcharge for a car service.

Does anyone know if you can sleep in the airport if they have pay showers or where would be the cheapest option to stay the night in Rome the night before

Google search:

https://www.sleepinginairports.net/sleep-guide/sleeping-in-rome-ciampino-airport.htm

The Rome Ciampino Airport closes every night from 12:00 AM to 4:30 AM, so it is not possible to sleep inside the airport overnight. If you’re already inside, airport security will likely ask you to leave.<<

There are lots of inexpensive hotels near CIA--start your search on booking.com
I would look for one that has an airport shuttle

The only way you can make the flight is if you take the 1:45 a.m. bus

That's what I'm seeing as well, not at "any hour of the morning". Voting with the rest that it doesn't make any sense to stay in Naples the night before this early morning flight. It's too big a risk to try and do so if it's important to you that you make your plane. Stay in Rome or Ciampino. If Ciampino, check booking.com for hotels near the airport that might offer free shuttles as well.

We stayed at the Sleep and Fly Rome B&B in Fumincino. Luca, the owner, will take people to the airport starting at 4:30 a.m. He charges 5 Euros for the trip. It was perfect for our 7:40 a.m. departure! We were told to arrive at the airport two hours early.

We stayed at the Sleep and Fly Rome B&B in Fumincino

The OP is flying out of Ciampino (CIA) and not Fiumicino.

Now way I would stay in Naples the night before.