ARD can work across the Internet as long as you make sure the necessary IP ports are open and forwarded in routers on both ends of the connection, and that you allow for changes to the IP address when the client system gets its IP address via DHCP. That can make using ARD across the Internet a pain, particularly when the client system is mobile.
I'm a bit confused as to which way you expect things to work. Are you planning to control and administer the Mac mini from your MacBook? Or were you thinking about administering the MacBook from the mini? You can actually do both, but the issue of changing DHCP addresses and network ports is more difficult when dealing with a mobile device as the client.
Just to make sure it's clear, the client software (already built into OS X) goes on the system you wish to control and administer. The ARD application you buy from the Mac App Store goes on the system you wish to administer from. This may be clear to you, but just to make sure.
Regards.