The entire Jewish nation, whom Jesus came to save, turned their back on Him. One of those closest to Him betrayed Him. Perhaps His strongest supporter denied knowing Him repeatedly. When He needed them most, His disciples couldn't stay awake. In a few minutes, Jesus knew all His friends would run away and abandon Him.
The cup represented the betrayal, the trial, the scourging, and the cross. He wasn't praying that God would let the cup pass from Him so that he could avoid all of those things. He was praying that He could drink so deeply from the cup that when he passed it on to humanity that the cup would be empty. Let me drain it completely, even the bitter dregs before I pass it on.