Fixing APC UPS buzzing noise
This is a YMMV / Caveat Emptor / DTTAH / ACNR type question and answer*:

ie the following may help but you MAY blow things up. 


  Is the buzzing because of the capacitor is bad or that a different component could be failing and having a secondary, audible effect on the capacitor?


Could be either. Capacitor is possible but I've not heard one do this. You may  get a better idea by using a 'sound transfer unit' eg a wooden ruler or length of solid plastic bar or similar. ENSURE it is not electrically conductive to the extent that applying 600 VDC on one end and your ear on the other will not cause "problems" [tm]. You may wish to use a piece of rubber glove on one end as well BUT in practice even touching the capacitor outer SHOULD be safe.
 Place one end of STU (sound transfer unit) against object to be tested and other end against ear or bone of head near ear. Sound can be localised and heard much better this way., Try on surrounding objects as well. Applying EHT to far end is to be frowned on - you should not have any EHT in a UPS.


  I thought the AC capacitors didn't have electrolytes in them and hence can't understand why they would be buzzing loudly when they did not previously


AC capacitors will probably contain two x electrolytic capacitors of twice the target capacitance each, connected in opposed polarity. 


  Should not both these caps be interchangeable in this application?


There is a "reasonable chance" that the cheap fan cap WILL work OK here. Also, used or dead fans may be cheaper again. 


  I am thinking of plugging out the 20uF 250V pedestal fan capacitor and putting this capacitor into such a fan instead, to confirm if the cap itself is causing the noise - sounds like a good idea?


Yes. Sounds good. Always a risk that the fan will stress it worse than the UPS does but if it is good it should work OK.


  I am not sure why the fans use 250V caps but my assumption is that these Chinese makers just stock up on one variety of the caps since most of the world are at or below 250V.


Many countries are 230 VAC. Some nominally 220 VAC. Voltages can be and are sometimes higher than nominal. 250 VAC is saying it will work anywhere.


  Hence, for this experiment, using a 150V cap instead of the 250V one for the fan (in the US) should not damage either fan or cap?


Probably. Cap MUST be AC rated and suitable for whatever class of service it sees. X rates is phase-phase or phase-neutral or line-line. Y rated is phase-ground. Such ratings allow for surges, spikes etc. 250 VAC cap in 110 VAC system should be fine.


  If it's confirmed that the cap's at fault, could I put in the 20uF 250V pedestal fan capacitor on the APC UPS board and expect it to have no detrimental effect on the UPS?


You can always EXPECT :-) ... .
 But, yes, it will probably be OK BUT no guarantee.


  My understanding is that this is a power cap and as long as the capacitance and voltage rating is good, other cap chars like ESR don't matter?


See above re X and Y rated. Also, some designs pass high currents at all times and some don't and a cheap cap may get sadder quicker. 



*YMMV / Caveat Emptor / DTTAH  / ACNR   = 
Your milage may vary
 Let the buyer beware.
 Don't try this at home (you can but ...)
 All care, no responsibility