The museums were free on the day we visited (perhaps because one of the main attractions - the art museum - was closed for renovation) though curiously we were asked repeatedly for our tickets in each museum of the castle! Admission prices are low anyway.
The ancient art section's worth a quick visit, with highlights being Leonardo's painted ceiling and Michelangelo's unfinished sculpture (Rondandini Pieta). There's a huge collection of musical instruments in that section - we didn't visit the furniture or prehistoric sections. But in many ways the biggest reason for visiting the castle is the building itself - courtyards, different styles of architecture and a large park at the back leading towards the Arch of Peace.
The castle's one of the most striking features of Milan, and it's worth an hour or two wandering through. Free entry to castle itself (small charge for museums). Worth a stroll through at least, but it won't keep you for a day.
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