A heliport was under construction on the roof of the Termini. You can prearrange over the web or book when you arrive at the Hotel Reservations desk outside the customs area. The most recent posting on the website puts the cost at just under a taxi, though recent feedback suggests a booking made on the spot at the airport is much cheaper, as little as half a taxi fare. Limousine Service Rome is one such. We've not used them, but have heard satisfied comments from those who have.
Prices appear very competitive, perhaps less than taxis.
They're connected, but it's much better to get off at the right one. If you have an early morning flight there's no need to stay at the airport. It takes only 30 minutes from most points, traffic is light and your hotel is accustomed to reliably ordering cabs for early departures. Nonetheless if you choose to ignore our advice Rome Airport Hotels provides a selection from which to choose. Questions about airport facilities are best answered at the Aeroporti di Roma website .