Hi Sarah,
It is possible to walk from the Old Town to the Strip and it would take about 20-30 minutes. Taxis are cheap though and should be easy to get in June and there's also a little
road train which runs all day until the early hours between the two areas. That website will give you a good idea of the general layout of the town. Albufeira is a hilly resort. The Old Town is down by the sea and you have to climb up to the main road which runs across the whole resort. Once you are up there, it's fairly flat across to the Strip, which then runs back towards the sea but only gently sloping until the end when you would have to go down "Cardiac Hill" to get down to Praia da Oura. It's hard to describe but I hope this helps!
There are lots of bars and restaurants in both areas and you will certainly be able to get a pizza or Chinese very late in the evening.