When Is the Best Time
You can enjoy a visit any season. They say that in the summer it's more beautiful. In general it's more pleasant to visit the city between spring and fall. During winter you don't experience a Siberian cold but the temperature still drops to a freezing daytime average of -12°C (10°F) in January/February.
Opening hours of the museum are: daily 11 AM - 6 PM (Saturday till 8 PM). Closed on Mondays. It can get a little busy if you arrive together with others on a tour but not heavily crowded. Visitors: 80,000 each year only.
Where and Tips
Yuzhno-Sakhalinsk was called "Toyohara" in those days. After the WW2, Soviet Union had begun to rule this area and the local authority used this building as Museum. This Museum is working till today. I feel that the Building itself can be something like an exhibition. These kind of unique style was popular in Japan in 1930's.
At anytime, you can see the people of this city enjoy walking around the museum. Official website in English: http://en.sakhalinmuseum.ru/ Information about the city on Wikipedia: https://en.wikipedia.org/wiki/Yuzhno-Sakhalinsk