The "cup of suffering" or "cup of judgment" or "cup of wrath" or "cup of trembling" refers to the same cup about which Christ prayed to be taken away while He was in the Garden of Gethsemane after the Last Supper. Instead, He drank it.
The idea was the "switch", the "covenant". At the Last Supper, Christ offered us His cup of the New Testament to the apostles who would then teach and do likewise (1 Cor. 11:25). In the Garden, the cup of trembling/wrath was drank to the dregs by Christ alone. One is either in that covenant or not. Do this in remembrance how that He died for our sins and offers us life.
Luke 22:42 Saying, Father, if thou be willing, remove this cup [of wrath] from me: nevertheless not my will, but thine, be done.
Isa. 51:22 Thus saith thy Lord the LORD, and thy God that pleadeth the cause of his people, Behold, I have taken out of thine hand the cup of trembling, even the dregs of the cup of my fury; thou shalt no more drink it again:
Jer. 25:51 For thus saith the LORD God of Israel unto me; Take the wine cup of this fury at my hand, and cause all the nations, to whom I send thee, to drink it.
Job 21:20 His eyes shall see his destruction, and he shall drink of the wrath of the Almighty.
Rev. 14:10 The same shall drink of the wine of the wrath of God, which is poured out without mixture into the cup of his indignation; and he shall be tormented with fire and brimstone in the presence of the holy angels, and in the presence of the Lamb:
The Catholic Church (CC) sort of makes the same connection.
612 The cup of the New Covenant, which Jesus anticipated when he offered himself at the Last Supper, is afterwards accepted by him from his Father's hands in his agony in the garden at Gethsemani,434 making himself "obedient unto death". Jesus prays: "My Father, if it be possible, let this cup pass from me. . ."435 Thus he expresses the horror that death represented for his human nature. Like ours, his human nature is destined for eternal life; but unlike ours, it is perfectly exempt from sin, the cause of death.436 Above all, his human nature has been assumed by the divine person of the "Author of life", the "Living One".437 By accepting in his human will that the Father's will be done, he accepts his death as redemptive, for "he himself bore our sins in his body on the tree."438