Limited traffic zone and having a car in Palermo

That is brave, if possible--apparently there is an app where this is theoretically possible, but I don't know it would be worth the hassle. https://wearepalermo.com/news/vehicle-parking-palermo/
I have never heard of it, but if you want t research it further, the app is out there (Italian skill would be helpful).
I would not dare consider this unless one has extensive driving experience in Italian cities.
I would instead split the trip and not use a car in Palermo, then stay somewhere easier with the car for exploring.

You should try to organize your trip in such a way that Palermo is at the beginning of your itinerary. That way you can be in Palermo without a car to see Palermo and Monreale (and possibly Cefalù, which is a short train ride away). Then after seeing Palermo, you can rent a car and visit other areas of Sicily with a rental car. You can also do the same by leaving the visit of Palermo at the end. In that case you would return the car when you arrive in Palermo, and stay the last few days in Palermo without the car. That is what I do, since having a car in Palermo is a royal PITA, because of traffic restrictions, traffic congestion, and parking availability issues.

If you must absolutely keep a car with you while in Palermo because Palermo is along the itinerary and not at the beginning or end of your itinerary, then a feasible option is to find accommodations which has parking or parking arrangements. That is easier if your accommodations are outside the ZTL. Some garages may offer that option also inside the ZTL, in which case they can list your car license plate on the white list for guests. However you will be limited to the gateways and roads you can use to reach the garage. Clients of hotels inside the ZTL can also obtain that temporary permit for a fee. Inquire with your hotel if you stay in a hotel that offers that option.

The Map of the ZTL of Palermo is accessible below.
https://www.accessibilitacentristorici.it/ztl/Sicily/palermo.html

There's lots to see in Palermo and the immediate area (though you don't necessarily want a car for those places), but plenty of Sicily's attractions are not within day-trip distance of Palermo, so I hope your aren't planning to spend all your time in Palermo unless your trip is quite short and you anticipate seeing only a rather small part of the island.

Traffic in Palermo is some of the most undisciplined I've ever seen. It's disconcerting to be a pedestrian there. I can't imagine driving in the city.

I still remember a woman in Palermo exclaiming to her partner as she backed out of a no traffic zone "but Goggle maps sent me this way!"

We rented a car for part of our trip too but not the part we were in Palermo. We returned our car at the Palermo airport and took the train into the city. Driving outside the urban areas was not difficult but I would not want to have driven in Palermo. It is in a class of its own.