Great stay in Palermo. The garden is beautiful, however it belongs to all the apartments. The apartment is close to the central station and right across the beautiful botanical garden. The other sights and most restaurants in Palermo are a bit further away, but all accessible by foot.
Palermo is a beautiful city, we found a lot of great tips at https://wearepalermo.com/, definitely try some of the local cheap restaurants and street food. We would skip the Zisa castle (all in Italian) and the Catacombs the next time, which are conveniently also about 45 minutes walking away.