Except private shuttle bus, is there any bus or other public transportations travelling from Český Krumlov to Brno? What is the best way?
Yes. You can take the Student Agency bus from Cesky Krumlov to Ceske Budejovice, then take another Student Agency bus from Ceske Budejovice to Brno. There's about an hour in between the buses, so you'll have plenty of time to get to your transfer. The bus station in Ceske Budejovice is quite nice, and attached to a mall, so you can go inside and have a nice shop or coffee or both while you wait. Go to jizdenky.studentagency.cz and book online.
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I believe that the "student agency" has busses doing that route.
It's a long ride though and I'm sure it's quite full. I'd recommend taking the train through Ceske Budejovice and Prague instead. It may take longer, but you'll have more 'personal space'.