Cheapest: Roma Ciampino airport -> Ciampino railway station -> Roma Anagnina subway station (€ 1)(Timetable). Then take the metro line A from Anagnina to Termini Station. Then change and take the metro line A from Termini Station to Coliseum station.
I don´t recommend...Terravision shuttle to Termini Station (8,00 € one way; 13,50 € round trip) (Timetable) taking the metro line B from Termini Station to Coliseum station (1€)
Couchsurfing
I'm a member of Couchsurfing, you can look for me on their website.
Some links
They might be useful... anyway just contact me and I will send you further information