No, the video data only goes through the mini display port (the small almost square one). I used to run two of these on a MacMini - to achieve that I need a converter from the mini display port to HDMI. They are not easy to be found and not very cheap. This one for example:
HDMI TO Mini DisplayPort adapter/ HDMI to Mini DP/ Ultra HD/ HDMI to Mini Display port converter/ HDMI to Mini DP adapter specifications : - HDTV resolution supported: up to 1080p@120hz, 4Kx2K@30Hz - Audio: 7. 1 Channel surround audio up to 192 kHz sample rated - compliant with VESA Dual-Mode Dis...
www.amazon.com
I guess once you have HDMI you can use a HDMI to USB-C converter to connect to your iMac. I will have my new iMac (hopefully) in a week and could give it a try if this works. But I fear it is time to retire the old Cinema Display.