If I buy a spectrum analyser, how do I do an EMI test?
Hopefully someone can provide a better answer as I only do a quick test before going to a proper EMI testing facility for a first pass.

What i do is use a scope / spectrum analyzer capable of accuracy up to the frequencies i'm concerned about, make a small coil of low gauge wire, and attach it to the end of a probe.  I hover this probe over various sections of the circuit that I had preemptive concerns about (and the rest of it) to look for "hot spots" and use some judgement regarding how hot they are vs past situation with similar circuits.

To ultimately "do it right" you need a spectrum analyzer, an antenna, and a "dead room" or testing box.

Depending on your continuing use of the equipment, the best answer may be to not buy the SA (unless you need it for something else too) and rather rent some time in a EMI testing facility, an hour or two with a EMC engineer will give you a very close to definitive answer as to whether or not you'll pass.  Unlimited hours without the proper facility will at best give you a "decent idea" of your chances.