If you have a spare 30-40 minutes walking thru old Alicante streets, you may want to visit Barrio Santa Cruz. In fact, that is a set of kinky pedestrian streets, sometimes less than 2 meters wide, mostly consisting of steps. The area is located on the hill side, so be ready to go up and down, almost no horizontal places.
Good idea to have a water and avoid mid-day to take a trip there. No special shoes required, aged persons or kids are not recommended to go.
Also, there is small modern church upwards (generally there are few of them around the area), and you may walk even higher to the Santa Barbara mount where small castle is located, as well as park with great views at Alicante and sea bay - add an hour to do that.More