I found, and have lunched at, a fantastic place right near Piazza di Spagna.
If you walk out of the metro to the Piazza, turn right, and then left at the first street you see, and you will see this on your left:
Every day, they offer two types of pasta you can buy to eat there. It is really more of a store than a restaurant, and I think that they make their money selling artisanal pasta and not meals. You can buy either fresh or dried pasta, but that is rather expensive. A meal, on the other hand, is only 4 euro. This does not get you anything fancy, just a simple plate of pasta.
And wine is gratis, although I think you are only supposed to take one cup.
Cold water is also provided