My ward in 23-ku also has English books in most all branches, so I guess it could be similar for many ward libraries, if not many public libraries nationwide. The online system is good for searching for and reserving books and having them brought into the branch that is most convenient.
Further to that, the wonderful Libby app (libbyapp.com) does work with library cards here in Japan, and most likely a library card from your home country as well. So you have not only free access to English books via your local Japanese library (and maybe your library in home home country too), but also magazines and audio books as well.