Great suggestions so far; I'll add one. Check out the Alpine Village Apts. in Tumwater on "T" Street. They have studios, my son lives there, and it seems like a decent complex based on the few times I've visited there. They don't show prices online so you'd have to call. That would give you at least an idea of rent and you can go from there. Tumwater is the town just south of Olympia and that location is a straight shot to Olympia's Downtown/Old Town/Capitol area.

I've never lived in Oly but have always enjoyed visiting. They recently built a waterfront boardwalk, and have a nice old town and farmers market. And it's close to nature!