If the start cap has died, and it still is tripping the mains then either the motor run cap has died ( very likely in these, as they tend to use a cheap one there) and you need a new one. If the capacitor is fine, and the only way to check is to do a capacitance check, and if it is not within 5% of marked value, and it will do an insulation resistance check pass ( 20M with 500VDC applied and no nasty crackling noises from the unit), and the start switch is fine ( a 40W incandescent lamp across the start cap) and opening as the motor runs up to speed then the only other likely fault is a shorted winding.
If the start switch is cactus then you can remove the start capacitor and buy a hard start kit for a compressor. This is a 200uf capacitor and a PTC thermistor, intended for a 6 cycles per hour duty cycle in an airconditioner. Does help with older compressors that get a little grumpy as they age, and for a motor with limited starts it also works well.