Can my new Mac mini M4 set-up with my old Apple Cinema HD 30" Display?

Yes, you should be able to hook up your Apple Cinema Display (dual‑link DVI, 2560×1600) to your mini, but it does require the correct adapters—and this is key: the M4 doesn’t output DVI directly, so you’ll need a USB‑C (Thunderbolt) to dual‑link DVI adapter that supports the full 2560×1600 resolution.

What You’ll Need

• Power the Cinema Display’s external power brick (it draws separate AC power).
• Use the Cinema Display’s proprietary dual‑link DVI converter box (comes built in with that monitor).
• Plug its USB-A cable into a USB port on the mini (needed for brightness, audio, camera).
• Then plug the DVI port from that box into a USB‑C (Thunderbolt) to dual‑link DVI adapter.
• Note that a simple HDMI→DVI or single‑link adapter won’t do—you’ll get only low resolution video.

I suggest that you search the Apple Communities to see what others have experienced with this type of setup. Other online forums can be beneficial here.

Note: Most common off‑the‑shelf adapters only support single‑link DVI (max 1920×1200 @60 Hz). But the Cinema HD 30″ needs dual‑link (2560×1600), so be sure you’re buying an adapter explicitly labeled dual‑link DVI‑D support up to 2560×1600.