No direct trains. You have to get the Leonardo da Vinci non stop train to Termini or the other train that stops at intermediate stations to either Trastevere or Ostiense and change from there. Takes between one and a half and one and three quarter hours. Ostia is not on the same line as Fiumicino Airport.
Try the Italian train site.
Trenitalia - HomePage
It only recognises Italian names so for FCO put in Fiumicino Aeroporto and do it after you click the Tutti I Treni tab. Le Fecce tab is for inter city fast trains only.