I think it would be easiest to hire a private driver for the drive from Positano to Napoli Capdichino Airport and then take a flight to Palermo with AirOne, Alitalia(watch out for strikes), or Alpieagles(choose Palermo in the drop-down menu):
You can also take an overnight boat from Napoli Molo Beverello or Salerno to Palermo:
Then I think you might need to take a train from Palermo to Taormina, however, perhaps someone might help out and let you know if there's air service available at least to Catania.
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