Sorry Amie brain fade here - confusing my towns. Yes the train from Fiumicino to the Termini will be the easiest and most direct route for you to take. Not sure you can use your eurail though as I have never tried. This is from a web site from googling - hope it helps. I'm sure someone esle will be able to help you through this site.
The train to Rome's Central Station (Roma Termini) runs frequently between 6:37 am and 10:52 pm and takes 31 minutes.
After you've picked up your luggage, follow the signs to Stazione FS/Railway Station.
You'll have to go up 2 floors to find it.
Buy a ticket "Per Termini" at the FS ticket counter or from one of the machines in the station. A ticket costs 11 Euros. You must validate your ticked by inserting it in the validation machine before you enter the train.