Yes it does sound like a bad run capacitor on the fan motor. It should cost around $5- $10. It should be about 5 MFD. It might be 4 or 7.5 MFD.
The compressor can be something in the 20 MFD to 40 MFD range, but I have seen 50 MFD on high efficiency units.
If the fan is not starting then the run capacitor for it is suspect.
IF the compressor is not starting (not what I am thinking in your case) then a super boost SPPE4 would help, especially in low voltage situations!
Good luck!
Fred.