Hi Lauren

You certainly can stay at many different convents in Rome for an extended period of time, I knew several people who did when I stayed there. My personal experience when I was looking for a place to stay though was that convents ended up being a lot of money for what you got overall. Basically, they're the same price as a room in an apartment, but you often share a bathroom with 5+ other girls, have a curfew you need to be in by, pay for all of your meals, even if you often wouldn't be there for lunchtime, and have to deal with lots of other people coming in and out who stay shorter periods of time and are quite loud. Many times the nuns don't speak any english either so you should know at least a little bit of Italian.
I hope I'm not discouraging you from the idea though! On the other hand, you would probably be able to find somewhere to stay for just a month easier if you were in a convent, and then sometimes the sisters allow access to their chapel as well. I guess it really depends on what you are looking for. If you plan on spending a lot of time in prayer and quiet reflection, you might prefer to live in a convent and might be able to make special arrangements with a particular community. If you want to primarily see and experience the city and its history, I would say getting a short term apartment might be a better route.
In general, expect to pay around 600 euros for one month's stay in a convent, including food, 400 without food. A room in an apartment runs anywhere from 200 euros to 1000+. I would say 200 euros a month for food and transportation, etc is plenty if you are frugal.
If you are interested in either the convent or the apt route, I have some contacts that I can give you. Living in Rome is a wonderful experience, and completely worth it, even if you aren't studying anything in particular!
God bless!
Claire