You’ve got a few possibilities here:
The first is setting up your old Mini to be a Server to your other Mini or other systems, then you can connect your two mini’s together via an Ethernet Hub.
The other option is to treat is more like an external drive by putting your older Mini into Target Disk Mode and then connect the two systems via a FireWire cable this will be faster than the Ethernet connection.
While these two approaches will get you over the hump if you need to get something done now, in the long haul it’s not very effective. How about getting an external drive and copy off all your stuff off then sell your old system off.