Palermo doesn't polish itself for visitors. It has been Phoenician, Carthaginian, Roman, Arab and Norman, and the last two left the best of it: in 2015 UNESCO listed nine Arab-Norman buildings in and around the city, from a palace chapel covered in gold mosaics to a Moorish-looking summer palace and a cathedral that was once a mosque. From 831 it was the capital of an Arab emirate, and from 1130 it was the capital of the Norman Kingdom of Sicily, which lasted until 1816. Then it spent a long time getting shabby, and a lot of it still shows.
That's the character: scaffolding, baroque churches, markets where a stall sells you a spleen sandwich, and traffic that ignores every rule. It's loud, it's hot for half the year and it's one of the best places in Italy to eat for very little money. Five days is the right length because the sights are small and close together, but the city punishes a rushed schedule: churches close for lunch, the markets are a morning thing and the early afternoon is for sitting down.
The plan keeps each day in one part of town. Day one is the Norman palace, the cathedral and Ballarò, day two the Capo market, Teatro Massimo, the Martorana and the Kalsa, day three the Catacombs, the Zisa and Monreale, day four a day trip to Cefalù, and day five gardens, the Politeama and the beach at Mondello, with an optional stop at the Falcone memorial tree. Start on a Monday if you can: the Royal Apartments in the palace are closed on Tuesdays and Wednesdays, and the Catacombs are quieter on a weekday. This guide stays in the west of Sicily. For Mount Etna and Taormina, see 5 days in Catania, and for the rest of the country start at Italy.