If your flight into Roma FCO arrives in the morning, there are DIRECT Sulga buses from FCO Terminal 3 to Perugia Piazza dei Partigiani at 9:00 a.m. daily except Sunday, and at 12:30 p.m. daily including Sunday. The direct bus takes three hours, 45 minutes to reach Perugia. Piazza dei Partigiani (bus depot) is within walking distance of many central Perugia hotels.
There are also direct Sulga buses to Perugia leaving from FCO later during the day (see hyperlinked website below).
Some travellers prefer the direct bus to Perugia from FCO Airport, rather than going into Roma Termini OR Tiburtina Station to take the Trenitalia train to Perugia Fontivegge Station.
If you fly into Florence FLR, there is an airport bus (or taxi) to SMN Train Station, then unreserved Trenitalia regionale trains to Perugia Fontivegge.
Ryanair flies from London Stansted Airport to Perugia.