After a fantastic 2 weeks in Sicily, I shall answer to myself in case someone is in the same situation:

The hotels inside the ZTL issue a pass upon check in. You give them your car’s registration plate and they will issue the permit online on the spot. This pass covers the day of check in and the day of check out. If you need to use your car in between, you will need to get another 5-Euro “pass giornaliero”, which can be bought online or at any tabacchi.

In order to access the ZTL on check in day we had to show our printed hotel reservation to a police officer who was standing just at the entrance of the ZTL. Once he saw the reservation, he was fine and let us go. Overall it was a pretty simple process.

Driving in Palermo is not as nightmarish as I had thought, overall it was similar to Buenos Aires but with narrower streets, although if you can avoid it, you best do. We only entered with the car because we were on a road-trip across the island and it was cheaper to keep the car for an extra day and leave it at the airport than to return it in the city.