transportation to airport

Yes.
http://www.sitbusshuttle.com/en/

Why the bus? Take the train from the St. Peters stop near the Vatican. I think the train is the FM1 and maybe 8 euro.

Frank, it doesn't look like the FM1/FL1 (same train depending on source) has a station near the Vatican? I believe they have to get it at Termini or Trastevere?

http://www.atac.roma.it/files/doc.asp?r=4

Looked around a bit online as I'm curious myself and am not seeing any direct train options from the Vatican area but I might be missing something?

Sam's option (SIT bus from Via Crescenzio no. 2, behind Castel Sant'Angelo) is probably the most direct method.

Yes, from Roma S. Pietro you have to change trains at Trastevere.