Would my monitor from my 2013 iMac work with a new Mac Mini?

No.

A Mac mini M1 2020 will not work with an iMac as an external monitor.

No Mac from 2020 and later will work with an iMac as an external monitor.

This independent of whether the iMac model supports Target Display Mode, which this iMac does.

Tap or click for the official details > Use your iMac as a display with target display mode - Apple Support (an article recently re-worded and un-retired, too.)

Here is the detail that blocks this configuration > The other Mac that you're connecting it to must have been introduced in 2019 or earlier and have macOS Catalina or earlier installed.

You need an external display for a new Mac Mini.

see > https://support.apple.com/guide/mac-mini/connect-a-display-apd8e4fbbb97/mac

I can verify all the previous posts as I have a late 2013 27" iMac

and it will not work in target display mode with an M1 Mac.