Hi
We did Rome on a budget last September, and it is certainly possible.
Save on bottled water by using the taps that are frequently located on the streets, it is perfectly safe to drink (I get holiday tummy, but I was fine, so it must be ok).
Don't buy any drinks from shops around the Trevi fountain, the prices are ridiculous!
You can pick up a cheap sandwich or pastry from the SPAR, there are various located in Rome. They sell cheap beer and wine too, and you can always sit and eat outside on a bench, rather than pay over the odds in a pavement cafe.
Do not eat on one of the main tourist squares, and pay attention to the prices, which should all be on display outside the restaurant. Generally if you go to pizzeria or trattoria they are cheaper than restaurants, and I would always recommend looking for places off the beaten track, not only are they cheaper but the food is often much better. You should be able to eat pizza or pasta for 4 or 5 Euros, and the local house wine is also usually reasonable.
Do not get taxis if poss, they are very expensive. If you must, make sure you agree the price in advance.
Lots of things are free to look at such as the Trevi fountain, Pantheon, circus maximus and spanish steps, other places charge, but its not a fortune.
Rome is walkable on foot, we never took any transport once we arrived in the centre, plus you get to see more by walking.
You will love it, and it can be as cheap or expensive as you make it, like anywhere I guess!
Hope that helps, and have fun!