I've professed my love of architectural museums, so it should come as no surprise that I took the path less traveled and made my way to Matsumoto's collection of old buildings.
The Matsumoto Japanese Court and Open Air Architectural Museum is open from 9:00 to 17:00, closed on Mondays. Admission is 400 yen. It's located next to the Ukiyo-e (woodblock print) Museum (10:00-17:00 except Mondays, 1050 yen), and for those interested in ukiyoe, that might be worth a stop too; I've heard that there is a relatively small number of prints on display at any given time, though.
Both museums are about 7 minutes by car from Matsumoto Station, and it's possible to take a taxi to the museum. I'm sure they'd call a taxi for you to get back, too. I went by train, which involves taking the Kamikochi Line of the Matsumoto Railway to Ohniwa Station and walking about 15 minutes. Keep in mind that Matsumoto is flat, so it's an easy walk. But you'll need a map, available at the tourist center in Matsumoto Station, and a good sense of direction. Ohniwa is a tiny station with nothing around it and no station master.