You should be able to connect the two using the Apple Thunderbolt 3-to-2 adapter. If the hard-wired cable on the Thunderbolt Display has gone bad, you may be able to compensate for that by running a replacement TB 1/2 cable from the Apple adapter to the Thunderbolt DIsplay's daisy-chaining port.
Don't use a USB-C to Mini DisplayPort adapter cable. The connectors would physically fit the Mini and monitor – but the cable wouldn't provide the TB signal the monitor requires. (Likewise, if this is actually a Apple 27-inch LED Cinema Display that expects Mini DisplayPort input, don't use the Apple Thunderbolt 3-to-2 adapter.)
That Thunderbolt Display has
- A resolution of 2560x1440 pixels
- (3) USB 2.0 ports
- (1) FireWire 800 port
- (1) Gigabit Ethernet port
- Thunderbolt 1 and MagSafe connections
The Thunderbolt 1 and MagSafe connections are obsolete – though you can work around this issue by (1) using an Apple TB3-to-2 adapter, and (2) not using the monitor to power or charge a laptop. Since you bought a Mac mini, you don't have any use for the charging power from the Thunderbolt Display, anyway.
The USB 2.0 ports are good for connecting keyboards, mice, even printers. They are not optimal for external drives and so you may want to connect those more directly to your Mini.
The FireWire port is a nice thing to have, if you've got old drives from which you need to recover data, old MiniDV or Digital 8 camcorders, or specialized audio gear for which people used to prefer FireWire to USB. If you do not have any of these things, you can ignore it.