There are lots of ways to accomplish this... the basic structure will be that you enable file sharing on the Mini and then make those ports accessible from the public internet. But there are some major caveats:
1) You will be exposed, so you really need to understand your security (e.g. if the Mini served a VPN, and all other ports were closed and stealthed, you would probably be reasonably secure).
2) You will need to know your IP, so if you are on a system where your IP periodically gets changed because you lose / regain your internet connection at the modem level, you're SOL
Otherwise, there is Mac software like the one described above....
The third option would be to just get a memory stick?

4 gigs, $30US... not bad....