The fact that no one responded to this point should answer your question. We recently sailed the Constellation with a stop in Palermo. We were so looking forward to Sicily since my wife's heritage is Sicilian. But we were totally disappointed in Palermo. Had I read up on the stop and been better prepared it might have made a difference. We just walked into and around the city from the ship. That was a mistake. Palermo is not a tourist town...it's a working class city with tons of traffic, tons of noise, and tons of smog. The streets in the old town are basically dark and narrow. No greenery or parks that we could find to escape the opressive pollution. We even saw people riding motorcycles wearing smog masks!
So what to do? We were only in port for 6 hours...so probably a ship's tour would have been better...although I didn't hear anyone rave about their tours. The cathedral at Montreale is supposed to be special...30 minutes outside of the city. We did find a beautiful cathedral in the main intersection of the old town but aside from that, leaving things to chance is not the way to go in Palermo. You need to read up on the stop (something I didn't do) and plan your time wisely.
We were in Palermo in May 2000 on a HAL cruise and we took the ship sponsored excursion to Solento. It was a very interesting tour of an area being excavated on a high hill. Was a Phoenician village at one time and the excavation shows what was probably a marketplace, theater, homes, and temple. Very good if you like old archeology. About a 30 minute bus ride up a hill and then some steep steps walking up there after you reach the bottom of the hill where there is a small visitor's center. Beautiful view of the Mediterranean from there too.
Agree with others that the town is full of smog, and narrow dark streets. We walked into town and visited a couple of pottery shops and bought a tea set. Not much more in the village.