If the mini supports analog monitor connections (and I don't know that it does), it should be possible but I'm not sure if you'll be able to just hook it up directly or if (more likely) it has a proprietary connector that will need to be adapted to a CRT connector.
That being said, there's two very, very good reasons for you to not do this:
1. The monitor in the iMac is, by modern standards, crap. Compared to a new display, the image quality is abysmal and the resolution is next to useless.
2. And the most important reason for not doing this... a crt monitor has voltages in it that usually work out to about 1kV per inch of diagonal screen size (edit: an EE friend of mine pointed out that it's actually more like 40kV, regardless of screen size- either way, it doesn't matter). When that machine is running the CRT is charged to somewhere between probably about 14,000 and 17,000v (or 40,000v...), and it can retain those voltages for weeks in some cases after it's turned off. If there's anything wrong with that monitor, or you make a mistake while you're in there it will. kill. you. And don't think "well I just won't touch anything in it", because electricity at that voltage is a completely different animal than electricity at 5v or 12v or whatever. 14kv will reach out and "grab" you-get your hand, a screwdriver, whatever-within a certain distance of a capacitor or something else charged to that voltage, even without touching it, and you're going to be doing a very good impression of half of a jacob's ladder. And something like that will just ruin your whole week.