How can I install Aperture on a Mac Mini?
Several options:


Remote Disk (using another drive over the network)
MacBook Air SuperDrive
other USB external drives
Pay again and download over the App Store.




The cheapest, possibly fastest is the first option:


MacBook Air, Mac mini: How to use Remote Disc to share DVDs or CDs from a Mac or Windows-based computer


The license code is non-transferable from physical media to the App Store, but perhaps you could sell your old version to someone that didn't like the store restrictions to subsidize or cover the cost?