In Mark 10:32, the evangelist writes that Jesus "took again the twelve and began to tell them what things should happen to him."
Jesus tells them,
Behold, we go up to Jerusalem, and the Son of Man shall be delivered to the chief priests and to the scribes, and they shall condemn him to death and deliver him to the Gentiles. And they shall mock him and scourge him, and they shall spit upon him and kill him. And the third day he shall resurrect. (Mark 10:33-35)
Then, James and John, the sons of Zebedee, come to Jesus, saying,
"Teacher, we want you to do for us whatever we ask." He said, "What do you want me to do for you?" They said to him, "Permit us to sit, one at your right hand and the other at your left hand, in your glory." (Mark 10:35-37)
To which request Jesus responds,
You don't know what you ask! Can you drink of the cup that I drink of, and be baptized with the baptism that I am baptized with? (Mark 10:38)
The sons of Zebedee respond affirmatively, and Jesus in turn responds to them,
You shall indeed drink of the cup that I drink of and be baptized with the baptism that I am baptized with! (Mark 10:39)
If indeed the cup was referring to the cup of the Eucharist, would Jesus have responded to the sons of Zebedee, saying, "You don't know what you ask! Can you drink of the cup that I drink of...?" For, certainly it was no difficult matter for the apostles to sit with the Lord Jesus Christ the night before his death and partake of the Last Supper.
But, as Heinrich Meyer stated, "The cup and baptism of Jesus represent martyrdom." Furthermore, Henry Alford wrote, "The one of these brethren was the first of the Apostles to drink the cup of suffering, and be baptized with the baptism of blood, Acts 12:1-2; the other had the longest experience among them of a life of trouble and persecution."
And so it was concerning this cup of agony and suffering, of martyrdom, that the Lord Jesus Christ fell on his face and prayed, saying,
O' my Father! If it be possible, let this cup pass from me! Nevertheless not as I will, but as You will! (Matt. 26:39)
And it was that same cup that the sons of Zebedee, as well as the other apostles, drunk when they experienced their own martyrdom or sufferings. But, not the cup of the Eucharist, or even an allusion to it, for the Lord Jesus Christ would not have asked (and did not ask) for that cup to be removed from him.