Im throwing up a few Rome posts from our December trip before my memory totally fades. Sorry for the limited specific food recs, but both of these are limited choice hard-to-go-wrong spots.
We have always had trouble finding good lunches around the Vatican - this year we went to two very good and reasonably priced casual spots which we highly recommend - both in Prati, which makes them particularly convenient before or after the Vatican Museums - both have a lot of character
Osteria dell' Angelo, Via G. Bettola 24 - a Slowfood spot - offer a very good fixed price lunch deal limited choices, roman food - they have an oven they fire up in the evenning but we were there at noon on Christmas Eve - patrons mostly local, groups of guys, families with strollers, all sorts - servers of character - supposedly ex-rugby players and looked it. Friendly and real.
Ditto L'Isola della Pizza at Via degli Scipioni, 43-47. Pizza at lunch is not all that common in Rome - this place offers fine wood oven pizzas along with a fine antipasti spread and I am sure other good things - Convivial good humored atmosphere, patrons from local business people to students - their pizzas were excellent. The owner/guy in front of the huge oven is reputed to be an avid hunter and game is featured in season - on a non-food note, funny sanitizing toilet is worth seeing - a new thing in Rome.