First, do not book for the night before, its a waste of money. The far better solution for jet lag is to change to the local time as fast as possible, by staying active and moving until as close as possible to normal bedtime in the new time zone. So walk around, see the sites, enjoy one of the best cities in the world.
Secondly, there are several threads here on hotels. Most like to stay in the historic center, but if your budget is a little more generous, I would stay at the Cavalleri, one of the best hotels in the city.
Lastly, early check in is always a function of availability. April in Rome isn't going to be as busy as other times...your chance of early check in is better. But hotels rarely refuse to hold your bags until your room is available. Plus, it will take you probably a minimum of 2 hours to clear out of FCO and get to your hotel, so you probably won't be there much before 10am.