Can I use a 2009 iMac as a monitor for a new Mac mini M4?

20" and 21.5" iMacs released in 2009 do not support Target Display Mode under any circumstances, and never supported it even when they were new.

24" and 27" iMacs released in 2009 theoretically support Target Display Mode using Mini DisplayPort input. If Apple still supported that, all you that would need to use it is a USB-C to Mini DisplayPort adapter cable (or the equivalent). Such adapters are inexpensive and readily available.

In the case of 27" Late 2009 and 27" Mid 2010 iMacs, Apple's own Technical Specifications explicitly say that "27-inch models also support input from external DisplayPort sources."

However, Apple later added restrictions on Target Display Mode. No current Mac supports using any iMac as a Target Display – both the hardware, and the versions of macOS that hardware runs, are too new.

See the bad news here: Use your iMac as a display with target display mode - Apple Support

No.

That’s not supported at all.

You’ll need a proper monitor to setup and use a Mac mini.

An iMac is a computer, not a monitor for another computer.

A quick search here in the community will produce many threads on the subject that include all the details.

Nope!

If you plan on getting an M4 Mac mini, then also plan on getting a decent external Display.

see > Connect a display to Mac mini - Apple Support

1440p (2560 by 1440 pixels) is barely past the resolution of a TV.

But if that resolution works for you though, new 1440p monitors are available for between USD$100 and USD$200. Cheaper used.