Your hint is right on the page you link to:
As NOAD says, one meaning of instant is:
instant (adj.) urgent; pressing: an instant desire to blame others when things go wrong
Synonyms of pressing include burning, as in:
There was a pressing/burning need to get supplies to the fort.
Furthermore, OneLook says that crying means "demanding attention", while Macmillan says that a crying need for something is an urgent need for something.
So, the synonym chain goes something like this:
instant > pressing > urgent > crying
I'm having trouble thinking of a sentence where I could swap crying for instant, but I can see how certain nuances of the two words are related.
As for your "clear" definition of the word synonym, it's important to highlight the word nearly:
a word .. that means exactly or nearly the same as another word or phrase
The word crying fits that bill, as both crying and instant can convey feelings of pressing urgency.