If you are lodging near Rome airport beware of extra charges at the Marriott Courtyard. They are now charging 6 euros per person for their shuttle to the airport, about 1 mile away. Taxis are expensive from the Courtyard and they are not near a metro or train stop, so they really have you with this surcharge. If you are staying there, add 24 euro for a couple to your room rate to cover your trip to and from the airport. i suggest Holiday Inn in Parco di Medici or one of the Hilton properties. They offer free shuttles and usually have equivalent room rates.
You can also take the Leonardo Express train from Termini to FCO. It's only 14€ each way. It is non-stop, which is nice. But you'll need to check the time table each way. I don't think the LE train runs before about 7am. Cabs are a flat rate of 48€ now. So if you're three or more, a cab is the best bet. Otherwise for 1 or 2, the train could be more economical.