Perugia airport is easily the handiest for Spoleto as it is only a 45 minute run down the superstrada however there are only a couple of flights a week (currently one a day on Monday, Wednesday, Friday and Sunday) with Ryanair from London Stansted.
As modemUK says the Rome airports are also convenient (not least of all because of the choice of flights) -both being about two and a quarter hours away. Rome is also probably easiest if you are relying on public transport (in particular if you fly into Rome Fiumicino you can get a train the whole way to Spoleto -changing only in Rome or Orte).
Although modemUK suggests Pisa, for Spoleto, Pisa is actually fairly far away (three and a half hours by car) and shouldn't even be considered if you are relying on public transport.
Another airport that is often overlooked is Ancona in Le Marche (again Ryanair from Stansted) which is about two and a half hours away by car. If you are relying on public transport there is a station just opposite the airport from which you can take the train into Ancona and can take a direct train arriving in Spoleto in around 2 hours.
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