It is not the cup that is poured out for us, but the blood of Christ. Your own translation, the Douay-Rheims, contains a most interesting future tense:
In like manner the chalice also, after he had supped, saying: This is the chalice, the new testament in my blood, which shall be shed for you.
The verb ἐκχέω ("pour out") together with αἷμα ("blood") invariably denotes sacrifice or murder in the LXX and the NT:
The rest of the bull's blood [αἷμα] he shall pour out [ἐκχεεῖ] at the base of the altar of burnt offering at the entrance to the tent of meeting. (Lev. 4:7)
Therefore this generation will be held responsible for the blood [αἷμα] of all the prophets that has been shed [ἐκκεχυμένον] since the beginning of the world. (Luke 11:50)
And when the blood [αἷμα] of your martyr Stephen was shed [ἐξεχύννετο], I stood there giving my approval and guarding the clothes of those who were killing him. (Acts 22:20)
Their feet are swift to shed blood [ἐκχέαι αἷμα]. (Rom. 3:15)
In other words, Jesus is using sacrificial language here, with an expression that implies death. And He was sacrificed on the cross, where His blood was poured out for our sake. The cup of the Last Supper points toward the cup of suffering that He prayed to be relieved of.