If you miss it, your ticket is dead the moment the train starts to move. However, four hours should be adequate but to be safe book the 12.45. What is the rush? That way you could grab a little lunch at Termini. Or walk out of Termini to get a little sunlight to boost the jet lag because you will be asleep on the train.
All trains in Italy except Regionale require a seat reservation that is specific to a train, date, and time. You cannot not just get on the next train without a seat reservation - the fine is huge if caught so don't even think about it.