Well, if you have password protection then I think all of the built-in abilities to reboot will require that you authenticate. I suppose you could have your password in a text file where you copy and paste it into a field, but you will need to authenticate.
The only other option I can come up with is to plug your mini into an IP addressable power supply and then you can bounce him by turning on/off the power plug (and yeah that might be a lot more complexity than you are looking for).
The Mac is now autobooting without a password, but I cannot get Vine Server to auto launch. I think it has something to do with permission and/or Keychain, but it will not launch until is authorize the System Server – which means using the keyboard.
Vine Server 3.0 has a small bug if it is set as a Login Item AND set to “hide”. This if fixed for 3.1 (and in the Beta if you want to download it) or you can just disable the Hide.