At Roma Fiumicino airport you can find the 3 H3G and Vodafone shops. All offer prepaid sim with data plan available as option. They require passport and codice fiscale, you can generate one online (just type generate codice fiscale in google with your details). Vodafone offer mobile internet at 3 Euro per week with a data cap at 500Mb per week, 3 offers superinternet is 5 Euro for 3GB of traffic valid 1 month, BUT only on 3 network, so when it roams internally to TIM you pay and also their network is not as extended as Vodafone, so I would choose the Vodafone.
Link is here with time open , is in Terminal 1
Shopping Gallery - Aeroporti di Roma - Rome Airports Shopping Gallery