The Sonobudoyo Museum in Yogyakarta
The Sonobudoyo Museum near the Kraton reputedly has a collection that is second only to the National Museum in Jakarta. There are some interesting displays, especially of shadow puppets, and there are some nice traditional architectural elements that you'd expect to see only in Bali, but the museum is very dark and many of the information cards are not in English (although at least some are).
At least one of the galleries was being remodeled when I was there in September of 2011, so an upgrade may be in the works.
Admission to the Sonobudoyo Museum is 5,000 Rupiah (0.30 USD).