The duty to assure that travel docs. are in order is 100% on the pax. Even if this were a UA-issued ticket on a UA-operated flight, it would still be OP's duty to present himself at check-in with the docs. required for the particular destination.
Given that the specific advice on UAE's website is to ask the carrier (Singapore) if it will arrange a 96-hour transit visa and OP clearly could not have done that, he's cooked.
Better advice to any prospective intl. traveler is to do the homework in advance. Carriers are sticklers about the rules because they are liable to fines and the duty to return you to your departure point if you are not properly documented and then not admitted.