Shikoku: Matsuyama Castle
The castle keep at the top and ninomaru near the bottom are both open 9:00-17:00. Admission to the keep is 510 yen, while the ninomaru is 100 yen. If you want to use the ropeway or chair lift, the fee is 510 yen round trip or 270 yen one way.
This was my last tourist stop in Matsuyama proper, but I'd recommend to others that you begin at Bansuiso and Saka no Ue no Kumo, visit the castle keep, walk down to the ninomaru, then end the loop at the art museum. This allows you to take advantage of the chair lift or cable car to get up, and avoid backtracking. Depending on how long you spend at each location, you can expect the entire loop to fill about half a day (I started around 10:30 and finished around 1:00). From there, you can head toward Dogo Onsen, some of the other museums or attractions in Matsuyama, or spend the afternoon at Kashima Island.