Himeji itself ended up being a bust. I didn't realize it until too late that Himeji Castle was under reconstruction, and thus covered by giant scaffolding. That is, by far, the most important reason to go to Himeji. There are other things to do in the area, but I figured if I returned for the castle I could do the other culture credits at that time too. So I skipped ahead and skipped over to Himeji Central Park.
Admission is kind of complicated. The safari bus is 800 yen and admission is 1550 yen. A free pass that doesn't include admission is 2800 yen. So a full, all-inclusive day at Himeji Central Park gets pretty pricey. There are all kinds of offers and packages and other details that make things more difficult if you want to add in the pool in summer or ice skating in winter.
Access is via bus #74 from Himeji Station, which runs three times a day on weekdays and hourly on weekends and holidays. Return buses follow a similar schedule. Ask for details at the Himeji Station tourist office.