How much per day???
One of the hardest things to figure out in Rome is your budget. How much should you budget a day? It all depends on your likes and dislikes. If you want a simple, quick coffee and roll for breakfast (and it doesn't come with the hotelroom), it will cost about 2 euros. If you want a full sit down breakfast, it will cost about 10 euros. If you want a slice of pizza and a soda to go, about 3 or 4 euros. A sit down, two course lunch, about 10 to 20 euros. A quick pasta and wine for dinner, about 25 euros. A nice sit down, full waiter service, four or five course meal with wine 60 to 75 euros. Remember, you will pay more to sit down. If you are at a pizza joint, just fold your pizza in half and take it to go. Stand outside and people watch or take it to a monument and have a seat. Also, don't forget about gelatos and sodas or water for the day. Gelatos cost anywhere from 2-4 euros depending on the size. You can buy a soda at a convenience store for about 1.2 euros. Water is about the same price but you can refill your water bottle from any fountain in Rome as long as there is no sign saying not to (have never seen one of those).
As far as entrance fees are concerned, if you budget 10 euros a sight, you will be okay. Some are more, most are less. Just figure out what you want to see and budget accordingly. Most sights in Rome are free. However, some things do require a fee. All museums, the colosseum, baths of caracalla, and the Borghese gallery all require a fee.
Souveniers are trickier. Rome is not a terribly expensive souvenier city. You can get some bargains. If you are looking for clothing, you can usually find the same thing for less back in the states. Same thing with good quality leather. Buy it before you go or after you get back. Any other type of souveniers, bargain for. Offer a reasonable price and you should be able to get it. If you want to take a day trip to Pompeii, the train will cost between 17 and 23 euros for a second class ticket each way. The circumvesuviana train is less than 5 euros. The entrance to pompeii is about 12 euros. You can buy food on the train or at the station outside of the ruins for a few euros. This is an all day trip, so bring some extra cash. You might want to do a guided tour of the site. ( not sure of the price for this but I would budget about 25 euros).
You can do Rome for 50 or less per day and have a great time or you can spend more and have a great time. I guess it all depends on where your comfort level is.