My wife and I will be arriving in Rome this Friday morning. We have ordered 2 Roma passes, which we will pick up from the Leo da Vinci airport. What is the best (free?) way for us to travel from the airport to our hotel? Our hotel is directly next to the subway station Barberini, but obviously sub stations do not go all the way out to the airport.
Not free but train to Termini the subway (subway is free with rome pass)
As David said, you won't be able to get into Rome free, but if you take the train from the airport to Termini train station, you can then use your newly acquired Roma Passes to take the metro to the Barberini stop.
There is no free way to get into the city unless you walk. The romapass does not cover ANY transportation to or from the airport.
You do have a few options. You can take the leonardo express train for 14 euros each. This will take you to the termini station. From there you can take the metro to the Barberini stop.
You can take the FM1 train to the Tiburtina station for 8.50. From there you switch to the metro going to the termini. If you plan to use your romapass this day you can use it on this part of the trip and also to get from the termini, where you change metro lines, to get to the piazza barberini.
You can take a cab for a fixed rate of 40 euros directly to your hotel or you can hire a company like romecabs.com to pick you up.