The apartment is across the street from Piazza Indipendenza which is a location where you can access a lot of public transportation; city buses, free buses, hop on/off bus, taxis, etc.
The train station is a 20 minute walk away. We made beautiful day trips via the train to Cefalu and Agrigento.
The traffic in Palermo seems chaotic and so it's good to get out of town for awhile.