How to connect a pretty old hard drive to MacMini?
You need an USB B to A cable.

USB B is the type of connector that fits your hard drive. 
USB A is what you'd call "classic USB" and that fits your Mac.



Alternatively you can use an Thunderbolt-to-Firewire adapter and then a Firewire cable. This might offer higher speeds. But it's also the more expensive option.