Early flight from Rome (Ciampiano)

a) Ask your Rome hotel to book a taxi to the air port
or
b) Stay at a hotel at the airport

Your hotel will be able to reserve a taxi for you, ask them to do it the night before you leave. I do this often.

I highly suggest using Romecabs.com. We have used them once for an early morning transfer from our hotel in Rome city center to the airport & once for a transfer from the airport to our apartment. We also had a 6:55 flight out of Rome this past September and the driver was waiting in the lobby of our hotel at the exact time that was promised. He is very friendly and helped us with our luggage. Both times they were on time and very reliable. It's about the same cost as a cab. You book ahead but don't pay until the morning of your ride. They always have glowing reviews and for good reason; they're very professional and reliable. Have a great trip!

I would set up a taxi through your hotel. Your other option would be to take a Terravision bus from Termini to Ciampino (the earliest leave Termini at 0430 and get to Ciampino at 0510, which is plenty of time for that small airport). At the same time - you'd probably be taking a taxi from your hotel to Termini, so you might just want to arrange it for the whole way to Ciampino.

Since you're going on a budget airline, have you paid attention to all the rules for baggage and such? I've taken many great bargain flights on Ryanair, but you just have to make sure you follow everything to a T. I found it easiest to pack my things in my backpack (school sized backpack, not backpacking around Europe backpack) because they seem much less likely to check it for size and weight, as opposed to suitcases. And when they say one bag, it means one single bag. You must fit your purse and such in your backpack.

Have a great trip! Ciampino doesn't have much to offer - and at least where I was there weren't many seats so we had to sit on the ground while waiting for our next flight. It's not luxurious, but it does get the job done.