You just share the HDD on your Mac mini and make sure it is mounted on the mac book air the first time you add it in time machine settings, thereafter, time machine will mount and unmount automatically when it does it's regular backup. You obviously need to keep the mini on if you want the time machine backup yo work.
You need Server for Time Machine backup over the network to another Mac. You might be able to get it working by sharing, but it is not a supported solution and will be unreliable.