Many times in the Bible, "drinking a cup" refers to the cup of wrath in the LORD's hand that he makes the wicked drink:
Psalms 75:8
For in the hand of the LORD there is a cup with foaming wine, well mixed, and he pours out from it, and all the wicked of the earth shall drain it down to the dregs.
Jeremiah 25:15-16
Thus the LORD, the God of Israel, said to me: “Take from my hand this cup of the wine of wrath, and make all the nations to whom I send you drink it. They shall drink and stagger and be crazed because of the sword that I am sending among them.”
For in the hand of the LORD there is a cup with foaming wine, well mixed, and he pours out from it, and all the wicked of the earth shall drain it down to the dregs.
Jeremiah 25:15-16
Thus the LORD, the God of Israel, said to me: “Take from my hand this cup of the wine of wrath, and make all the nations to whom I send you drink it. They shall drink and stagger and be crazed because of the sword that I am sending among them.”
Jeremiah 51:6-8
“Flee from the midst of Babylon; let every one save his life! Be not cut off in her punishment, for this is the time of the LORD'S vengeance, the repayment he is rendering her. Babylon was a golden cup in the LORD'S hand, making all the earth drunken; the nations drank of her wine; therefore the nations went mad. Suddenly Babylon has fallen and been broken; wail for her!"
Isaiah 51:20-22
Your sons have fainted; they lie at the head of every street like an antelope in a net; they are full of the wrath of the LORD, the rebuke of your God. Therefore hear this, you who are afflicted, who are drunk, but not with wine: Thus says your Lord, the LORD, your God who pleads the cause of his people: “Behold, I have taken from your hand the cup of staggering; the bowl of my wrath you shall drink no more…"
Revelation 14:9-10
And another angel, a third, followed them, saying with a loud voice, “If anyone worships the beast and its image and receives a mark on his forehead or on his hand, he also will drink the wine of God's wrath, poured full strength into the cup of his anger, and he will be tormented with fire and sulfur in the presence of the holy angels and in the presence of the Lamb."
Revelation 16:19
The great city was split into three parts, and the cities of the nations fell, and God remembered Babylon the great, to make her drain the cup of the wine of the fury of his wrath.
Surely the reference to the "cup" in John 18 also refers to the cup of God's wrath, which Jesus was about to willingly drink from the Father's hand, the wine of God's fury poured out upon himself in place of his people who were wicked:
John 18:11
So Jesus said to Peter, “Put your sword into its sheath; shall I not drink the cup that the Father has given me?”
The cup of the Lord's Table is his blood. Why? Because by drinking the cup of God's wrath, the Savior died. The cup we drink symbolizes the cup that Jesus drank for us. We remember the cup of wrath, from which we have been saved by his drinking of it. We remember that what it cost for him to save us from the "wrath to come" was his drinking it himself until the cup was drained.
Matthew 26:27-28
And he took a cup, and when he had given thanks he gave it to them, saying, “Drink of it, all of you, for this is my blood of the covenant, which is poured out for many for the forgiveness of sins."
1 Corinthians 11:26
For as often as you eat this bread and drink the cup, you proclaim the Lord's death until he comes.
1 Corinthians 10:16
The cup of blessing that we bless, is it not a participation in the blood of Christ? The bread that we break, is it not a participation in the body of Christ?
1 Thessalonians 1:10
…Jesus who delivers us from the wrath to come.
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Christ also speaks of the "cup he must drink", in Mark 10:37-40; there it seems to refer in general to a type of suffering which all of his disciples will share in if they will also share his glory (Rom.8:17). But the overall symbolism in the Bible refers to God's wrath against the wicked.
