One thing to beware in wintertime: traditional Japanese houses are designed to be cool in summer, which all too often means that they are freezing cold inside in winter.
Unlike in undergraduate programs, however, it is less common for graduate students to take outside their specific field of study at graduate or level.
It is common to visit shrines and temples to offer coins and make silent prayers, and many homes often have a small shrine or display religious artifacts from various temples, but there is little mention of religious faith or dogma in daily life.
If you do manage to get in, don't let your washcloth touch the water, as it's considered mildly bad form; you may wish to fold it atop your head, or just lay it aside.
In its feudal days, you could find multiple castles in nearly every prefecture.
Just make sure you reserve your ticket in advance.