Free bicycles from “Sui Sui Town” are typically available daily from 9 a.m. to 5 p.m. and can be returned as late as 9 p.m. at various locations throughout town.
Information is available in English on the Matsumoto City website (at the bottom of the page).

The other free bicycles known as ‘Rikisha’ are also available at some hotels.

The bicycles are convenient for visiting sights not accessible by Town Sneaker Bus—you will not win the Tour de France on one, but you can get your souvenirs in the front basket. Details are also available at Matsumoto Tourist Information Centers.
Additional information available via this blog article about rickshaws and free bikes.

Safety

As Japan drives on the left side of the road, bicycles should also be ridden on the left side. Many of the streets around Matsumoto Station have bike lanes, and you should make every effort to ride in the road with traffic when possible. Don’t hesitate to use your bike bell and be on the lookout for elderly cyclists or pedestrians suddenly cutting in front of you!

What to do if there’s an accident: In event of an accident, drivers are obliged to contact the police & bow at you until they get a nosebleed/get a mortgage to pay you off. In all seriousness, apologizing can land you in hot water, so try to speak directly to the police only. Hit-and-runs do occur. Try to get the number of the car and insist on medical attention. Dial 110 for police and 119 for an ambulance.