I mentioned recently that I went back to Meiji Mura (link is to my first trip's post), the largest and best architectural museum/historical village in Japan. With a heavily-discounted admission, I wanted to explore some buildings a second time to retake photos and take additional angles. And, of course, I got to sample the local beers and a delicious snack.
That isn't true of any of the buildings, though - they are all open and free to be explored in part or whole. Just expect to take your shoes off.
Abashiri it is not, but you could peer in a few cells and enter a couple too.
You'd think that two visits are enough, but if I ever have the opportunity to stay even longer I would enjoy participating in the tours, eating more of the food, trying out the transportation, and exploring even more of the buildings in detail. They hold seasonal events, too, and I'm sure this place looks awesome in the fall and in the snow.
I posted details on the museum in my first post, but times change, so this will bring it up-to-date. You can verify details via the museum's English website.
The park's hours are 9:00-17:00 March through October, 9:00-16:00 November through February. Note that the museum closes several days in December and January (especially Mondays).
Regular admission is 1700 yen, but discount options are available; mine was literally given to me, so I don't know how to obtain them. I believe combination options through rail or bus companies are a good place to start looking, or ask at visitor's centers in Nagoya or Inuyama.
Access is best via a bus that leaves from Inuyama Station; it departs about once per hour and English signage at the stop is pretty good. Get off at the end of the line; you probably won't miss it as the bus deliberately pulls into a pickup/drop-off area at the front gate.