JR "Azusa" limited express trains require just over 2.5 hours for the journey between Tokyo's Shinjuku Station and Matsumoto. The one way trip costs 6620 yen and is covered by the Japan Rail Pass and JR East Nagano Niigata Area Pass. There are one or two trains per hour.
Take the Hokuriku Shinkansen from Tokyo to Nagano (80 minutes, multiple trains/hour) and transfer to a JR "Shinano" limited express to Matsumoto (50 minutes, one train/hour). With a good connection, the entire one way trip takes around 2.5 hours, costs around 10,000 yen and is covered by the Japan Rail Pass and JR East Nagano Niigata Area Pass.
Highway buses operate hourly between Tokyo (Shinjuku Bus Terminal) and Matsumoto (Matsumoto Bus Terminal near Matsumoto Station). The one way trip takes just over three hours and costs 4300 yen. Discounts apply when tickets are purchased online. Online reservations can be made through Willer.
Above fees and schedules are subject to change. Be sure to check current yen exchange rates.
Central Matsumoto can be explored on foot. The castle is about a 15 minute walk from Matsumoto Station. Alternatively, bicycles can be rented from multiple locations across the city through the Hello Cycling app. Unfortunately, the app is difficult to use because it is only partially in English. The former free bicycle rental system is no longer in operation.
The Town Sneakers are three tourist-oriented loop bus lines, which cost 200 yen per ride or 500 yen per day (the one day pass also includes a 10% discount on admission to Matsumoto Castle). Buses on the loop lines operate approximately every 20-30 minutes (the south line hourly). Besides the "Town Sneaker" buses, there also exists a regular network of city buses.
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