As an Airbnb host myself, I learned a lot about hosting from staying here!! Marcello and the apartment are accurately reflected in the many very positive reviews here. I agree with them; indeed the apartment is as advertised and great, they are wonderful hosts, their recommendations are excellent, and the place is great with a baby (the high chair was excellent for our 9-month-old).
I can only add this: Monteverde (the neighborhood of the apartment) is not within the ancient Aurelian Walls (from the 3rd century or so), so not technically the center, but it is right next to them. Even better: it is on a hill (the Janiculum) that is high and has great breezes, which really make a difference in the hotter months. It's a "real" neighborhood, not full of tourist schlock. I've stayed in Rome a dozen or so times, including one stint of a year; I will always return to Monteverde because it is convenient, quiet, real, and very comfortable. If you want the noise, heat, and crowds of the absolute center, then don't stay here, but if you want a place to relax and experience a better part of Rome while taking advantage of the center, then stay in Monteverde. I certainly will -- hopefully often in this apartment!