Nagano Olympic Museum at M-Wave
The Olympic Museum is open only on weekends 10:00-17:00, though the souvenir shop is also open on weekdays. Admission is free but there is essentially no English information. You should be able to figure most of it out just by looking at it, though you miss out on the story, history, and names.
To get to the museum by bus from Nagano Station, go to Bus Stop 1 and take a bus toward Suzaka & Yashima, getting off at the M-Wave-Mae bus stop, which takes about 15 minutes. The return bus stop is across the street. There isn't really anything else around M-Wave, so plan on returning to Nagano afterwards.