Matsumoto has a small local airport, with commercial flights only from Osaka, Sapporo, and Fukuoka. Travel time to downtown is roughly 20 minutes.
From the closest major airport, Chubu Centrair International Airport near Nagoya, Matsumoto can be reached by rail in roughly three hours by the Meitetsu line and JR limited express train (change in Nagoya). Buses connect Centrair with Matsumoto in around 3 1/2 to 4 hours for ¥3500.
From Narita Airport, Matsumoto can be reached in 4 1/2 to 5 hours by limited express rail service (change in Shinjuku).
Matsumoto is located on the JR Shinonoi Line. It is four stops away from Shiojiri, the "mid-point" of the Chuo Main Line between Tokyo and Nagoya.
Limited express trains run to Matsumoto from Tokyo's Shinjuku station. The Super Azusa (スーパーあずさ) makes the run in about 2 1/2 hours; the Azusa (あずさ) makes the journey in about three hours (¥6710 for both).
Taking several local JR trains will cost roughly ¥4000, getting you from Shinjuku to Matsumoto in around six hours... in which case, you're better off using a bus (see below).
There's also the option of taking the Nagano ShinkansenAsama (あさま) from Tokyo to Nagano station, then taking the Wide View Shinano (ワイドビューしなの) limited express to Matsumoto (2 1/2 hours, ¥8470).
The Wide View Shinano also runs from Nagoya station, where there are connections with the Shinkansen line (towards Kyoto and Osaka). The run from Nagoya to Matsumoto takes 2 hours on the limited express and costs ¥6070; local trains cost ¥3260 and take between 4 and 5 hours, depending on the available trains and connections.
All of the above mentioned rail trips are free when using the Japan Rail Pass.
Matsumoto Castle (松本城 Matsumoto-jō). Completed in 1614, this stately landmark is considered one of Japan's Top 3 Castles. Visitors can climb up precarious stairways all the way to the top, and the castle also houses a comprehensive exhibit on the history of firearms in Japan. Located 15 min north of Matsumoto station on foot, admission ¥600, open 8:30 AM to 4:30 PM daily. Tickets include admission to the Matsumoto City Museum on the castle grounds, with artifacts documenting the history of the city.
Nawate-dōri (なわて通り) is an old-fashioned shopping street running along the north bank of the river, full of small shops selling antiques, foods, used books, etc. The old movie theater shut its doors after its run of "The Last Samurai", but the posters are still up as well as the nostalgic building itself. The street's ubiquitous mascot is a frog, orginating from frogs in the river as well as a pun on the Japanese word for "return" 'kaeru' (the mountains could be treacherous, frogs were given as a charm so that loved ones would return safely. It was also hoped that money and goods would also return.)
Matsumoto is mildly famous for its soba noodles, hopefully served with a dab of product from the world's largest wasabi farm nearby.