Two ATMs and exchange counters that you can use. I know both work with the Plus network and Visa cards. Make sure your card has a 4-digit numeric PIN, as European ATMs usually only take 4-digit PINs and don't have letters on the keys.

First is immediately after immigration on the right. The second is between Arrivals areas "B" and "C". Since you will be arriving at "C", exit Customs and turn left. There is a Banca di Roma branch about 50 yards on your left, just before your reach Arrivals area "B". The ATMs are on the wall just around the corner.

One caution about "never bringing any Euros". Due to an idiotic policy change at FCO, it is the only international airport I've ever flown into that charges for baggage trolley use. The carts cost 1 Euro (50 cent or one-Euro coins only) and there are no change facilities in Arrivals A or B. I've never tried to get change at the exchange counter in C. Best advice is to make sure you have a one-Euro coin with you if you want to use a trolley.