A train to Perugia requires a high speed train to Florence and a Regionale Veloce to Perugia. About 5 hour journey.
The full walk in price in standard class is about 75€ pp one way. If you buy a few weeks in advance you might be able to save over 25€ per ticket.
Fontignano is also not far from Chiusi, however all rental agencies, Chiusi or Perugia, are closed on Sundays, except for the Perugia airport.
Personally, since you plan (need) to rent a car on that day, I would pick up the car in Venice and drive the 5 hours. It will be cheaper than a train if there are 2 or more people in the party.
Not all agencies are open at Piazzale Roma (Venice) on Sundays, so you might need to pick up the car at the Venice airport.