really not my area of expertise, but I can offer these few pieces of information.
You will probably be looking to take a train from Rome to the airport, it is about a 40-minute ride. The train departs from Termini, and this is a large train station (and wouldn't you know it the train to the airport is all the way at the end of the tracks). There is also security control block before you enter to the train platforms - so give yourself plenty of time to navigate the station and I suggest that you have your tickts purchased before hand (you should be able to do that online).
You could also hire a car to take you to the airport, make sure the prices are negotiated before you leave, and enquire if luggage costs are extra. (let me know if you want the name of a reputable service)
As per where to stay, I personally have stayed several times near the train station ( https://www.hotelwindrose.com/en/
) however, I was always there specifically to catch the train the next day - and never took the time to walk around Rome for "one last memory". In fact, staying around the train station is probably known as one of the more friendlier areas to walk around at night due to the excess of neon lights, kitsch souvenier shops and international dining opportunities.
So, if your last night in Rome idea is to enjoy the lights of Rome, the fountains, Spanish Steps or the Vatican, then you may really want to consider something closer to the city center, or nearby one of the more popular monuments, and then plan on using a driver or taxi service to get you to the airport (hassle free, and no navigating the train station)