All four hotels are within a few minutes of the terminals by car. The difference is how you get there.
The walk
Hilton Rome Airport. No bus, no timetable, and a 24-hour front desk, per Booking.com. It costs the most.
The free shuttle
Hilton Garden Inn. ADR runs the Airport Line from Entrance 3 of T3 arrivals, and says it runs every day, 24 hours, every 15 minutes. One older listing says 05:00 to 01:00, so confirm for an early flight.
The paid shuttles
Best Western's is reported at about EUR 8 per person each way, first run 04:30 and last back 22:30. Mercure's is on request at about EUR 6 per person each way, until 22:00. For two people on a return trip that is EUR 32 and EUR 24, on top of the room.
Compare on our budget short list and the early-flight short list.