FCO -> Naples, take the 9 am bus. You will have no issues with the GRA.
Naples -> Rome (city). Take the train, unless travelling in August. The worst part is usually from the GRA to the city centre, but since everybody is away from Rome in Augustall big roads (Tiburtina, Nomentana, Tuscolana, Salaria etc.) are empty and it's really fast and easy: I've done it in less than 10-15 minutes and even found a parking spot in front of la Sapienza at around 11 am Rome in August is the ideal city for cars...