If your stay in Rome is hardly two to three weeks long and you are intereseted in sight seeing at a reasonable price, you don't need to buy a monthly bus pass. You can travel anywhere in Rome throughout the day just by purchasing the 'metrebus' ticket for 1Euro each. You can change as many buses as you want and you are also allowed one subway (underground railway) journey during that period, using the same ticket as there aren't any zones to limit you from traveling from one end of Rome to the other (unlike London). Tickets can be bought from all tobacco shops which have the blue signs 'Tabacheria'. Moreover, there are night buses as well which travel specfically in the evening.
Aeroporto-Roma Cornelia (metro A) (€2.80)
Aeroporto-Roma Magliana (metro B) (€1.60)
Aeroporto-Ostia Lido (€1.00)
Aeroporto-Fiumicino (città) (€0.77)