Quick way to find magnetic nature of compounds or ions
Well, for some compounds it's easy, because there are an odd number of electrons - thus there's no way to pair them all. 

Otherwise, there is no quick-fire way without doing the MO diagram.

Consider $O_2$ which has an even number of electrons, but is a ground-state triplet with two unpaired electrons. The only way to know is to look at the MO diagram.

If you practice, you'll realize certain patterns and develop more of an intuition.