Shutdown MacMini with hardware button
For a clean shutdown (the button solution is not clean and there is no way around it, i am afraid) i suggest you to connect to your iTunes Server via ssh from a second machine and shut it down this way.

Another option would be through Apple Script. If you have a specific time that you want to shutdown your iTunes Server, or if there is no request inbound to this server for a specific time a script could be auto-activated to archive this.

Though the first option is much easier to accomplish!