This apartment is about as close to Trevi fountain as you can get! It was wonderful to wander down at 3 am and see almost no one around. You can also peek out the window in the apartment and on the floor and see the fountain. About a 15 minute walk from the Colosseum and Forum, 10 minute the other way to the Pantheon.
Cendra was very detailed about how to find the apartment and gave great advice on how to get there using public transit or private taxi.
The apartment is small, but well suited to a pair of travelers. It has a kitchen, eating area, washer, bathroom, and lovely A/C. Cendra had even provided water and snacks as well as some basic toiletries although there's a store just downstairs in case you need to buy anything.
Because it's right near the fountain, you do hear some slight noise from the crowds, but mostly the fountain just sounds like a white noise machine.
I don't think you can beat sitting by the almost empty fountain around 4 am, so set an alarm or stay out late!