On busy holiday weekends you might find a chap in traditional Nagano jogging gear + chariot at the castle. Shinshu Matsumoto Rickshaw provides basic information on their facebook page and offers courses starting at ¥1,000 (¥1,800 for two people) for 15 to 20 minutes. Longer courses are also available.
Available for special bookings, too. (Wedding photos at the castle anybody?)