Here's some mobile offers from Italian carrier TIM aimed only at foreigners (non-Italians):
TIM International Pass
Euro 12 / 10 Days
500 minutes calls
500 SMS
2GB Internet on 4G
TIM International 1000
Euro 10 / month
1000 minutes calls to anyone
3GB Internet on 4G
Unlimited calls from 1 cent per minute to fixed lines and mobiles from Italy to abroad.
Free activation
TIM International 2GB
Euro 3 / 4 weeks
2GB / month on 4G
Free activation
TIM International 10GB
Euro 15 / month
Internet up to 42.2 Mbps
You will need to show your passport and maybe even your Tax File Number to benefit from these offers. Italians usually show their ID cards and Codice Fiscale (Tax File Number) if requested.
I've only listed some plans. There are several others for foreigners and a few more for Italians that foreigners can sign up to. Best to see them personally to work out which plan is best. Many of the sales people in the bigger shops speak English. So there shouldn't be any communication problems. If you arrive in Rome and are taking the Leonardo Express (train) to Termini Station you'll find the TIM shop on the upper level above the Trenitalia ticketing counters. I've been there many times to reactivate my account. Access is by stairs and escalator. Another phone provider, Wind, has a shop on ground level off the concourse. Not sure what offers they have. Also Vodafone may have similar offers. They have a shop on underground level. Provider 3 also have a shop down there. These last three providers I have not checked their plans. But can do if requested.
See this interactive map of the Termini Station:
Maps - Roma Termini map - Station map shops | Roma Termini for the locations of the shops.
The Leonardo Express stops at platform 24. On the far right looking at the trains from the concourse.
Should you require technical assistance, such as a repair or even advice in English, I can suggest a reliable and trustworthy technician I met several years ago when I first activated my account with TIM. Just let me know.
Cheers