High speed train from Orvieto to Fiumicino airport

The high-speed train (Frecce) does go to the airport, but only twice a day. The problem you have is that it the high-speed trains don't stop in Orvieto. If you want to take the train that morning, you'll need to take one train to Rome (either Termini or Tiburtina) then a train from there to FCO. Takes over 2 hours--assuming your 11:10 flight is non-stop out of the Schengen Zone, you should be at the airport 3 hours before scheduled departure. Looks like you'll need to be boarding a train in Orvieto at 05:54 at the latest.
If it were me, I'd move the day or evening before to Rome.

Where is the 11:10 flight going to? You need to arrive at FCO about 90 min prior for a European flight and if the flight is bound to North America, you should try to get at the FCO airport about 2.5 hours prior to flight departure.
There are no high speed trains from Orvieto, you need two trains. One from Orvieto to Roma Tiburtina or Roma Termini, then one other train from the aforementioned stations to the airport.
See what works for you at that time in the morning.
www.trenitalia.com
Station names are:
ORVIETO
FIUMICINO AEROPORTO

There are no direct trains going from Orvieto to FCO. There’s a train leaving Orvieto at 6:13 that arrives at 8:18. You would have to changes trains at Roma Tiburtina. Next train leaves at 7:25 and you change to the Leonardo Express at Termini arriving at the airport at 9:07. Weekend schedules may differ.

Sounds to me like a good reason to leave Orvieto on a late-afternoon or evening train and overnight in Rome. But I really hate stress rushing.

I am with Laura---- Spending the night in Orvieto with a 11:10 flight is a recipe for di$a$ter.

Take a late train to Rome the night before.

Traveling any distance longer than Roma to FCO to make an international flight is not a cogent strategy. It's really a bad idea because too many things can go wrong causing you to miss the flight. If it were me, I would spend the last night in Roma and use the Leonardo, a taxi, or a private car to transit to FCO. Generally, we arrive for our international flight 3 hours ahead of departure to insure our connection. Your choice.

Buon viaggio,

Just adding my voice to the choir: it's unwise to spend the night that far away from the airport before an international morning flight. It increases the risk of things that could go to heck in a handbasket. Spend the night in Rome.

You could even consider to spend your last night at the airport, if you have seen Roma before.
The Hilton is conveniently located across from the terminal. The cost is about $250 with tax, but considering that you save time and €50 taxi fare in the morning, it is not much more than staying in a hotel downtown Rome.