So you have at least 2 and possibly three faults.
There is a quick and dirty way of checking the capacitor on small air compressor motors, if you have no test gear and no way of getting it easily.
With the machine unplugged, remove the end guard off the motor, and if possible remove the fan. If the fan comes off, you then plug the thing back in and turn it on. Then give the shaft a quick flick, with the intention of pulling your hand away as you twist. If the motor starts, its the capacitor.
This is not as dangerous as it sounds, because the round shaft is smooth and there are no edges to catch your fingers. I've done this, and it works. 8)
If the fan wont come off, then you need other testing means because that IS risky to turn by hand. :shock:
Then you just need to look for the air leak.