Usually, if the iMac lost power suddenly (e.g. pulling the AC plug from power while it's still switched on) you would need to allow some time for the capacitors to discharge before working on it. However, if the iMac was shut down properly (e.g. Apple > Shut Down) then it should discharge the power supply capacitor fairly quickly (not immediately, but close).
Typically, the input capacitors on a computer power supply have between a 180 to 360 volt charge on them. How long they hold this charge depends on the circuit, but a bleeder resistor sits across them to dissipate this charge.
Regardless, in your case it sounds as if the iMac hasn't been powered up for a while and the power supply isn't working. Of course, since it's broken (or faulty) there's always the potential for a problem. And, since you can't be 100% sure what the problem is, make sure there is no AC cable plugged in while working on it.
TIP: DO NOT wear a watch, bracelet, necklace, etc (or any metallic objects) which could accidentally make contact with the power supply circuitry.
Regarding the use of a multimeter, you could test across the capacitors with one. If you actually get a voltage reading then you could even use the multimeter in this way to discharge the capacitors slowly until the voltage reads zero. If you're wanting to use a multimeter to discharge, I'd recommend an analogue one as a high impedance digital meter would take a while to discharge.
However, as I mentioned above, it sounds as if the iMac hasn't been powered up for a while, so you should be right to go.