During the day it is possible to get from Fiumicino Airport into town center by train.
At night (from about 23:30 until about 6am) the only options are Taxi, NCC (cars with drivers) and buses. For those on a budget buses are probably the best opti0n.
The more reliable up to date information is on the offical site of the Rome airport company, at:
here they also tell you where each Bus leaves.
Public transport run by ATAC (Rome public transport compay) is priced cheaply compared to other capital cities, so, for instance, it is possible to go to the sea with a one way ticket that costs 1.50 euros (Ostia, the part of Rome on the sea it's still ... Rome, despite being 15-20 Km away from the town center. Note that the nightly buses that travel between Fiumicino airport and Rome town center do NOT belong to ATAC (the airport is in not in the town of Rome, it's in Fumicino, another town) but are private except one run by COTRAL, the public regional transportation company; so this is rather cheap: 7 euros in 2017 and even though the buses look old and unattractive they usually have good airconditioning and ... do the job in a good way, sometimes with pragmatic flxibility on the time-table (waiting some minutes to leave if there aren't many people). You can buy COTRAL Bus ticket on the bus, from the driver.