Both terminals have currency exchange desks and ATMs, so arriving without euros isn't a problem — though where you exchange money and how much you convert makes a real difference to what you actually get back.
Where to find them
Currency exchange desks and ATMs are available in Terminal 1 and Terminal 3, in both landside and airside areas.
A tip on rates
Airport exchange counters generally offer weaker rates than a city bank or your home bank's card — if you only need enough cash for the taxi or train into Rome, exchange a small amount at the airport and do larger exchanges (or just use a card) once you're in the city.
Which banks operate ATMs here
ATMs at the airport are generally operated by major Italian banking groups alongside independent currency-exchange operators, similar to what you'd find at any large international hub. Withdrawing from a bank-branded ATM rather than a standalone currency-exchange kiosk's machine typically comes with a more transparent, lower markup on the exchange rate.
Card payments are widely accepted
Contactless card payment is standard across shops, restaurants, and even the taxi rank and train ticket machines at Fiumicino, so carrying a large amount of cash isn't strictly necessary for most of a typical arrival. Notify your bank of international travel before you fly if you're concerned about a card being flagged for an unfamiliar country.