Radya Pustaka Museum is the local historical museum for Solo. In it's many small rooms are displayed some fine examples of shadow puppets, keris knives and other cultural artifacts. The displays are rather old fashioned and dusty, and there are no signs in English, so unless you're a real museum buff I'd suggest giving the place a miss.
Admission to the museum is 5,000 Rupiah (0.30 USD). There's an additional 5,000 Rupiah fee for a camera if you want to take pictures. It is open Tuesday to Sundays from 8:30 am until 1:00 pm.