Can I use my old iMac as a screen for my MacAir?

If your MacBook Air is running Sonoma as shown in the profile information you posted, then the answer is no. Not unless you can find a way to downgrade it to Catalina or earlier, and comply with the remainder of those instructions for the old iMac (reinstall High Sierra or earlier).

No. See: Use your iMac as a display with target display mode - Apple Support

No, Target Display Mode does not work on recent computers.

Shortest user tip ever:

NO!

Apple already has a support article, though I note that my Mic Mode User Tip complements a support article, too.

We can get a clip of Marcel Marceau saying "No!" in Silent Movie.

Given the frequency with which this question appears, someone really ought to create a User Tip on the subject.

👍

I like it.