Can a 2009 Apple Cinema Display be connected to an early Mac mini?

If the "hydra" cable has a USB head, you're definitely not looking at a 27" Apple Thunderbolt Display. It didn't need a separate USB upstream connection.

As far as video goes,

• Mac minis released between Early 2009 and Late 2014 all had Mini DIsplayPorts or Thunderbolt 1/2 ports which could act as Mini DisplayPorts. Provided that the Mini in question supported the monitor's resolution, you would plug the Mini DIsplayPort head of the "hydra" cable into the Mini's Mini DIsplayPort or Thunderbolt 1/2 port.
• Mac minis released in 2018 and later have USB-C / Thunderbolt 3 or 4 ports. For these, you could get a USB-C (male) to Mini DIsplayPort (female) adapter. You'd plug the monitor into the adapter, and the adapter into the Mini.

You'll probably also need to connect the USB uplink if you want to be able to control the brightness of the monitor, or use any downstream hub ports it might have.

I have this monitor connected to an Mac mini (Early 2009)

also works with a Mac mini (Late 2009) and (Mid 2010)

Apple Cinema Display (2009) | MacRumors Forums

it says it has displayport connector (it's the same connector as thunderbolt 1 and 2) you can get an adapter if it's the under thunderbolt 3 which use usb-c connector