The Tokaido Hiroshige Museum of Art focuses on Utagawa Hiroshige, the great ukiyo-e artist best known for his prints of the 53 stations of the Tokaido road. Several of those stations were in what is now Shizuoka, making the museum a natural stop for travelers interested in the old highway.
The collection is strongest if you already care about Japanese prints, Edo-period travel, or the way the Tokaido shaped the region between Tokyo and Kyoto.