I’m not familiar with AirBnb in Rome but for 150 euros per night you should be able to find a reasonable flat in a nice area (Pantheon, Trastevere, Prati, etc.). It really depends on you, but I’d much rather spend my time there than in the area near Termini, which is quite bad — every large city has a rough neighbourhood, and Termini is central Rome’s.

We go to Rome at least once a year and usually stay at a nice “residenza” on Largo Argentina which charges between 70 and 80 euros per night for two people, breakfast included. I don’t know if that type of hotel suits you, but it may be worth to pay a look. Sometimes they have bigger rooms that can accommodate more people.

How much is the bus from FCO to Termini? Rome Shuttle Limousine, also recommended here, charges 50 euros and they have large vans that can accommodate up to 6 people + their luggage. I have always found that public transport from FCO is always convenient when travelling alone, otherwise the difference is negligible and the limousine will be waiting for you and will drive you straight to your hotel.