I was in Sicily the last part of Sept & early Oct a few years ago. We stayed at an AirBnB a few blocks south of Teatro Politiana, an area with many small grocery stores, restaurants and neighborhood feel. As 2 woman of a 'certain age' I never felt unsafe walking around, even after dark.
Cefalu is an easy train ride from and back to Palermo. We had lunch, visited the Norman Cathedral (unbelievable mosaics), walked to the water and did some shopping/browsing. My concern for you is that your free time is over a major religious holiday, which Sicilians take very seriously. Public transportation will probably be very limited.
Palermo is very walkable. Besides what has been previously suggested, I would add Parco Piersanti Mattarella (English Garden). It is located on Via della Liberta, north of Teatro Politiana (less than 1 mile walk). It's a beautiful garden with fountains, benches and many locals. Across the street is another garden (less lush) with statutes etc. You can Goggle to see if this would interest you.
I hope you love Sicily!!!