You write that the motor works fine when you help it start with a nudge. Also that you can start it in the "wrong" direction. And it does not start on its own (mostly). This is indication for a capacitor which is either dead or which has lost a lot of its capacitance.
With the limited tools you have got, there is mainly one test that you can do: Purchase a replacement capacitor and install it. If it starts the motor relyably every single time, then you can conclude that the old capacitor was the problem, and you have already fixed your water cooler (yeah).
I just had a similar problem, with the extra problem on top that shipping to West Africa often takes several weeks or even months and is costly. So I learnt on this forum that even a difference of 17% in capacitance can still start your motor.
If you are reluctant to purchase a new capacitor (I am guessing, because you are posing a question here, rather than just ordering one)(it should not be too expensive in relation to your entire cooling machine), look around your workshop and see what other capacitors you have available. Or even borrow from another working appliance. Maybe you find a 10 uF, just give it a try. I propose to not go more than 20% over the rating of the original unit, i.e. not over 14 uF.
I noticed your comment about testing with mains voltage. This is what I have observed as the "normal method of testing" here in West Africa (by craftsmen who repair motors but have no proper testing tools). It is spectacular and will indeed give you some information, if you manage to have the spark and the bang. You might need several attempts (maybe 20) as mains is of course AC. You need to very quickly disconnect after charging. And I will not give you lectures about handling mains and capacitors. But we suspect that your capacitor is dead and you cannot prove that with the spark-bang-test.
Anyway, John Birckhead has already written nice testing instructions. Testing for "not working" is harder than testing a properly working unit. So again, if you have any other capacity available, you can double-check if you have a good set-up and how a working capacitor behaves. hth