The Bishop's Castle (Püspökvár) is the oldest surviving structure in Győr, its thick medieval walls enclosing Chapter Hill on the north side. The complex today houses the Diocesan Treasury and the Lapidarium, a collection of Roman stonework from the ancient settlement of Arrabona that once occupied this site.
The castle tower offers one of the better elevated views over the city and the river junction below. You can walk through parts of the fortifications; the exhibition inside covers the castle's history from Roman times through the Ottoman sieges to its current role as the bishop's residence. It's not a major museum, but the tower climb and the medieval stonework make it worth an hour.