The flat is just south and a short walk from the Old Town (a little uphill on the way back). The apartment is spotless and quiet with all you'll need. The host, Valentin is very communicative and provides plenty of suggestions for places to visit and eat.
The Old Town holds much interest and a walk along the river to the Guggenheim and Modern Art Museum is a pleasure (lots of joggers and dog walkers). For a vegetarian, though, Bilbao's pintxos offer little choice.