There is a Terravision coach specificaly for Ryanair, easyJet and some other budget airline passengers from Rome Ciampino to Termini Station in Rome. It takes 40 minutes and costs 8 Euros per person.
It leaves from the Arrivals carpark and passengers have to show a valid airline boarding pass, ticket or e-ticket for one of the appropriate airlines in order to board the coach. They depart quite regularly and there's one timetabled for 12.30pm.
Tickets can be purchased onlone on the Ryanair website.
From Termini to your hotel would be a lot cheaper than taxi the whole way. There is no train link.