If you search for a hotel inside Rome airport, the honest answer is that there is not one you can sleep in inside the terminal.
What ADR lists
ADR's hotel page names three: HelloSky Air Rooms, Hilton Rome Airport and Hilton Garden Inn. Only the Hilton is walkable to the terminals. It is joined by a covered tunnel with a moving walkway, and ADR says it takes under 10 minutes on foot.
HelloSky Air Rooms
ADR describes an arrival lounge with rooms you can book from a few hours to several days. A 2023 third-party guide places it on the second floor between Terminal 1 and Terminal 3 and describes daytime rooms of 3 to 6 hours. It is reported closed from 1 October 2026 until 20 January 2027, which we have not confirmed. It is not on Booking.com, so we do not list it as a stay.
The next best thing
Hilton Garden Inn has a free ADR shuttle that ADR says runs 24 hours every 15 minutes. See the short lists for Terminals 1 and 3.