Hotel in Taormina shuttle to Catania airport

I have not but it's a very easy and inexpensive bus ride (luggage goes on the side of the bus). The bus station is a 5 minute walk from the main town gate. I'm sure you can find a hotel that will do a private transfer - many hotels in Taormina are luxury hotels so they'll have no problem with the request. You can also search the hotel listings on www.booking.com by those that identify shuttles as an amenity.

In April 2017 we stayed at Hotel Villa Angela. The hotel is not in Taormina town itself, but in the hills overlooking the town, very close to Madonna de la Roca. We arranged with the hotel for a private transfer from Catania airport as we were coming off a trans-Atlantic flight from Texas with a change of planes in Europe. This is from the booking confirmation email:

"By Air

From the Catania “Fontanarossa” airport, you can reach Villa Angela in different ways: - by taking the public coach to the bus terminal of Taormina (the journey lasts one hour and 15 minutes). Here, you will find a taxi service that will take you to the hotel for approximately € 15.00. - take a private taxi directly to Villa Angela, reserving directly through our hotel. The price for this service is € 85.00. - get a rental car and take highway A18 toward Messina. Exit at the Taormina tollboth and continue on to the city center following signs to Parcheggio Porta Catania. At the roundabout, follow signs to Castelmola. Our hotel is located approximately 1 Kilometer ahead, below the Sanctuary of Madonna Della Rocca."