Quite the opposite of the way Klinghoffer views Paul's interaction with the Old Testament, he claims that Jews (like himself), "...do just the reverse. They prioritize the Old over the New. They begin with the premise that God speaks through the Hebrew Bible. With this in mind, they may proceed to evaluate the claims of Christianity. Because they approach and interpret scripture in chronological order - as seems reasonable, after all - they find that the New Testament does not arise naturally or logically from the foundation document, the Old Testament." - p110, Why the Jews Rejected Jesus.
First of all, of course they prioritize the Old over the New. In fact, they reject the New outright. But his point is deeper than this. He believes that Paul, and Christians ever since, reinterpret the Old with the erroneous New. Interestingly, this is just what Jesus was dealing with in the Sermon on the Mount. Over and over again, He says, "You have heard it said....but I tell you...." Many, including Christians, take this to mean that Jesus was ramping up the law, or reinterpreting the law. But that is not what He was doing. He was reorienting the listeners to the original point, the original interpretation and intent and application, of the law.
Second, throughout the New Testament, the writers show their logical and chronological handling of the Old and New - Klinghoffer is missing this. Hebrews 1:1-2, for instance: "God, who at various times and in various ways spoke in time past to the fathers by the prophets, has in these last days spoken to us by His Son, whom He has appointed heir of all things, through whom also He made the worlds..."
The New Testament never teaches an abandonment of the Old Testament. It does regularly teach an abandonment of damnable misinterpretations of the Old Testament and false-teachers who mishandled it to the detriment of their followers: "Woe to you, scribes and Pharisees, hypocrites! For you shut up the kingdom of heaven against men; for you neither go in yourselves, nor do you allow those who are entering to go in." - Matt 23:13.
The New Testament in many ways is a testimony of Israel's unfaithfulness to the Old Testament. It is a renewal of covenant with those who would respond to God's Word with faith, like their spiritual father, Abraham. And Klinghoffer cannot see that because he calls himself a son of Abraham while not having the faith of Abraham. He cannot see that his people executed another Prophet, just like they did countless other times - prophets whose books they now include in their holy book.
"Woe to you, scribes and Pharisees, hypocrites! Because you build the tombs of the prophets and adorn the monuments of the righteous, and say, 'If we had lived in the days of our fathers, we would not have been partakers with them in the blood of the prophets.' Therefore you are witnesses against yourselves that you are sons of those who murdered the prophets." - Matt 23:19-31