I found this page of "free things to do in Rome" while trying to find the answer to if the Vatican is free during cultural week. I haven't found an answer to that yet but the last Sunday of each month the museums are free, also there is a "white" night in September that museums are open all night and waive the fee. Wow, that would take care of your staying awake if you arrive late afternoon and want to stay up for get acclimated to the time.
http://www.italylogue.com/things-to-do/free-things-to-do-in-rome.html
I love Googling subjects. I find so many other interesting information.