The cup from the Passover Seder to which you refer is the Elijah cup, placed on the table for Elijah who is the herald of the Messiah (Mal. 3:23-24). Christians believe that this prophecy was fulfilled by John the Baptist, herald of Jesus Christ (Matt. 17:9-13).
We don’t know if the eucharistic cup of the Last Supper was the Elijah cup or not. After the destruction of the Temple in Jerusalem in A.D. 70, the rabbis had to work many years to standardize Jewish faith and practice without a Temple. Without primary source documentation, it is difficult to determine if the Passover liturgy of the first century is the same as the Passover liturgy of rabbinic Judaism. Thus, we don’t know if an Elijah cup was used in the Passover liturgy during Jesus’ time. And, even if that could be determined, it is difficult to determine from the Gospels which cup Jesus used as the eucharistic cup during the Last Supper.
