Kekayon Museum is a wayang puppet museum set in a lush Javanese-style garden southeast of Kotagede. Its galleries are divided into sections, with puppets from different parts of Indonesia and displays that explain the forms and stories behind them.
It is a specialist museum, but a worthwhile one if you are interested in Javanese performance traditions. Combine it with Kotagede's old lanes and silver workshops rather than crossing town for this alone.