Smell it first. Look at it, see if the windings are discolored or burnt. These are the most subjective tests, so it takes some interpretation, and there are no guarantees that a motor with darkened winding insulation is "toast".
The most common fixable problems with small motors are dirt accumulation leading to overheating, bearing trouble/lack of lubrication, defective capacitor, and defective switch on start winding. The first will result in intermittent operation while the others will prevent starting under load or maybe result in intermittent operation.
It takes a voltmeter or test lamp to make sure the motor is actually getting power, and supposed to be running. If the motor is powered, but not running, FIRST disconnect the electricity from the machine before doing any other testing.
Spin it by hand to check for bearing or other trouble. Check for high temperature of the motor. Normal temperature is too hot to keep your hand on it for long, but not burning hot.
Then check the capacitor connections, look at the capacitor(s)for swelling of the can, burned connections, leaks etc. If you can, check the capacitor with a capacitance meter, if you cannot do that use an ohmeter set on a high (kohms)scale and after disconnecting the leads, short a metal object across the two terminals to discharge the capacitor, and then touch it with the ohmeter probes. A bad capacitor will either show infinite resistance or very low resistance that stays low, where a good one will charge up and the resistance will start out low and quickly climb toward infinity. This is only an indication, if you suspect a capacitor problem, replace it with a good one.
On a single phase motor with or without a capacitor, there may be an internal centrifugal switch to cut the start winding out when the motor gets up to speed. This is not common on newer blower motors any longer. If the centrifugal switch fails, the motor will have to be opened up to fix the problem. Usually it is not cost effective to have a repair shop do these kinds of repairs on a small motor. If you want to examine the motor internals, sometimes the problem can be fixed simply or a switch replaced.
The most perplexing motors are the ones which check OK on all the simple checks, but still operate only a short while or fail to start etc. The megger testing would probably reveal turn to turn winding shorts or other insulation failure in at least some of these, but those are truly uneconomical to repair these days.
You will find examining old electric machinery an interesting passtime and sometimes what you learn will save you some money. Sometimes though it's best to get a new one and get the AC back on.
BE SAFE
Jim