How can I use 2013 iMac as a display for my Mac mini m4?

You cannot.

That's a dead end.

The iMac was not designed to be used as a monitor and the tech that had enabled that for a minute died with the release of retina Macs in 2014 and macOS Big Sur years ago.

A decent monitor for the mini will cost $100-$350. You should budget for that when you buy the new Mac if you don't already have another monitor.

Not possible and there are no workarounds, hacks or magic powders to make it work. I know this as a fact as I own a late 2013 iMac.

If you are willing to tear it apart and gut it, search the internet for solutions.

No, the new M4 Mac mini does not support using an iMac as a display in Target Display Mode.

Use your iMac as a display with target display mode - Apple Support

If you have or get a new M4 Mac mini, then you will need an External Display or HDMI TV.

Connect a display to Mac mini - Apple Support

Using an iMac as a monitor - Apple Community