Just to add to what ChristineH is saying, the train, either from Rome or from Florence, is totally doable. On our trip throughout Italy this past fall, we used the Trainline app extensively. You enter the departure and arrival locations, and it gives you all the trains that will get you there, as well as identifying the fastest and cheapest options. And you can buy your train tickets right there on the app.
Florence (which appears to not be your preference for flying in), happens to have a direct train to the Cortona Camucia station. This might or might not work with your schedule, but as an example, this is train #RV 4077, departing 12:14pm and arriving at that particular Cortona station (there are two stations for Cortona, as you’ve probably ascertained) at 1:34pm, an hour and 20 minutes later, for less than $13.00 US.
Rome takes longer, as it’s farther away, One train involves two transfers, both in the Rome area, and one transfer gives you 6 minutes to make the transfer, and the other 11 minutes. You don’t want to dawdle, but it won’t require an Olympic Gold Medal sprint effort, either. And this one was under $16.00 US. Your particular train itinerary could differ, but it won’t be an impossible task.
The hardest thing, at least for us, was the bus to get up to the Cortona old town, up the long, steep hill. If you’re in the newer, lower town, that won’t be such an issue. Or if somebody picks you up at the station. Otherwise, buy your bus tickets from the bar at the station, go to the bus stop just outside the building, and then let the driver know where you’re going. Then hold onto your luggage as the bus climbs, climbs, climbs. The driver let us know when we’d reached our destination.
