The pagoda-roofed Sakhalin Regional Museum has a strong exhibit (sadly in Russian only) exploring the Japanese/Soviet overlap of the city’s history, typified by the building itself, which served as the home of the Karafuto administration before the Soviets seized the island from the Japanese in 1945. The strong ethnographic section has some fascinating and unique Aino artefacts and photos from back before the original south Sakhalin inhabitants fled to Japan, plus bits on the Nivkhi and the rare Aleuts.
The gorgeous gardens that surround the museum are worth a wander themselves, and include various Japanese items captured during the war, including artillery guns and a tiny tank.