Arriving in Trapani on May 27 midday and leaving very early on June 1. We will have a rental car. Doubt we will go to Palermo. Considering the time constraints, what are must sees and any room recommendations?
I recently did a similar-length trip in Sicily, but was on the other side of the island. We did take a day to check out Monreale and Cefalù, in terms of things on that side of the island. Monreale is on the west side of Palermo and I would highly recommend you go there. The church is stunning and it is a charming hilltop town. Cefalù is roughly 50 miles east of there. It is a very charming coastal town. In terms of church stuff, the Cathedral is incredibly disappointing. It does have a very fine altar that is quite famous, but apart from it and the apse mosaic, the rest of the church is wholly underwhelming. But the piazza in front of the Cathedral is nice, especially for having a drink and people watching, and like I said, the rest of the town is quite charming. If you do take any day trips east, I highly recommend Syracuse, particularly the little island that the Cathedral is on. Taormina is also well worth seeing. There are a lot of great things to see in Sicily, it just depends on what types of things you want to see, and how far you want to go on this trip. You may want to just concentrate on one area and, having whet your palate, go back again in the future to see the other great things that it has to offer.
Agree with Bryan, Monreale is great but I would also say Palermo is a must see. If you don't want to drive into Palermo (and you don't) an option would be to drive to Falcone/Borsellino airport, park the car and catch the Prestia bus into town. Takes about an hour, 5.80 per ticket and they have 8/9 stops throughout town. Buses run from around 5 in the morning to late at night (9/10 pm. not quite sure offhand but can check if you like). With the time you have I would say stick to the western half of the island. Trapani, Marsala, Agrigento, Segesta,Erice, Monreale and Palermo will provide plenty for the time you have.
There's a nice place on the Marsala/Trapani line, about a 10 min drive from Trapani airport. Baglio Oneto. Nicely furnished, good food, pool, reasonably priced, good views and surrounded by a vinyard and olive trees.
Just a few kilometers from Trapani is the hilltop town of Erice. The Greek ruins of Segesta are not far away, either. Monreale, definitely. I love Palermo and Cefalu is one of my favorite places in Sicily. I don't think you'd have time for more in 4.5 days. Unless you want to squeeze in Agrigento (a long daytrip from Trapani).