It's another illustration of the non-infallibility of the foototes in Catholic Bibles. (See the January 1994 "Dragnet" column for an instance in which we nailed another incorrect footnote.) While one might suggest that this prediction is a paraphrase of something Jesus said rather than an exact quotation from him, it must be regarded as the substance of one of his actual historical utterances.
This is underscored by the fact the Gospel writer gives a specific time when Jesus began to make this claim ("From that time on . . ."). It was not a bit of embellishment invented by a later writer and inserted to give a literary flourish. It was something Jesus actually said.
The only reason anyone ever challenges the idea that Jesus predicted his passion, death, and resurrection is out of an anti-supernatural bias. Jesus could not have predicted these things because, the reasoning goes, that would mean he knew the future, which is impossible. The idea that Jesus never predicted his own death and resurrection became popular over a century ago with liberal Protestant Bible scholars, and they infected many Catholic Bible scholars in turn.
This attempt to de-supernaturalize the consciousness of Christ was recently refuted in the Catechism of the Catholic Church, which states,
[The] truly human knowledge of God's Son expressed the divine life of his person. The human nature of God's Son, not by itself but by its union with the Word, knew and showed forth in itself everything that pertains to God. . . . The Son in his human knowledge also showed the divine penetration he had into the secret thoughts of human hearts. By its union to the divine wisdom in the person of the Word incarnate, Christ enjoyed in his human knowledge the fullness of understanding of the eternal plans he had come to reveal. (CCC 473-474)
Thus Christ humanly knew the supernatural mission he had come to perform and what it would involve.
The Catechism also deals with the texts where Christ predicts his Passion and Resurrection:
"When the days were near for him to be taken up [Jesus] set his face to go to Jerusalem" [Luke 9:51]. By this decision he indicated that he was going up to Jerusalem prepared to die there. Three times he had announced his Passion and Resurrection; now, heading toward Jerusalem, Jesus says: "It cannot be that a prophet should perish away from Jerusalem" [Luke 13:33]. (CCC 557)
The footnote in your New American Bible is thus not only completely out of line with the historical teaching of the Catholic Church, but with the Church's contemporary teaching as well.