"To drink from a bitter cup" usually means to submit/surrender to an unpleasant feeling or 'continue to do something unpleasant', even though you know what's going to happen. I believe the term comes from Jesus Christ sharing cups of wine with his disciples, even though he knew that they would not speak for his defence at his trial, nor stand up for him on the cross. Defoe wrote books to educate and warn the people on the best way to avoid the plague, but knew that it would probably come to London anyway. You could interpret "that bitter cup has come out of his cupboard" as people already knowing Defoe was correct, the plague was coming, but pretending it wasn't and going about their daily lives.