I just returned from a 3 week trip to Italy. I purchased a voucher from TIM on their website. TIM for Visitors Card is valid for 30 days and includes 4 gigs of data, 100 minutes of calling and a €5 credit for €30. I redeemed the voucher at the TIM store at FCO and it took about 15 minutes for the employee to inserte the card into my phone and activate the service. The store is located in the Boarding Area B concourse before leaving the airside so if you have a connecting flight, you don’t have to go through security.
Service was great in the areas where I travelled - Sicily and Orvieto south to Paestum. On rare occasions coverage dropped from 4G to 3G. The plan does not include SMS, so messages are charged at 15€/cents each - charged against the €5 credit. I used the phone constantly for research, mapping, GPS, streaming (not too much), email, social media, calls. In the time I was there, I usd 3.5 gigs of the data allowance and still had a €4 credit left.
If you are arriving at FCO, you’ll have your phone up and running before you leave the terminal.
EDIT: I landed in terminal 3 but the TIM store is actually in boarding area B in terminal 1. I didn’t realize that the terminals were interconnected on the airside. Just follow the signs to B and you’ll be fine - but it can be a loooong walk.