The ancient roots of Scholasticism were Aristotle and Augustine. Centuries after both of them were dead, the philosophers of the Middle Ages created the logical style of analysis which we call Scholasticism, and which formed the foundation for modern physics, mathematics, and chemistry. Scholasticism was characterized by rational debate, in which various viewpoints were examined carefully. As a movement, it reached its high point by around 1250 AD, and was on its way out by the 1400's. It lay dormant during the Renaissance era, when there was neither interest in logical debate, nor openness to competing viewpoints. Scholasticism was re-incarnated as modern philosophy in the disputes between Descartes and John Locke, and in the innovations of Isaac Newton and Robert Boyle.
But can we say that Augustine himself was a Scholastic? Usually he is identified as a "root" of Scholasticism, as one who laid the foundation for it, but not as a Scholastic proper.
Augustine was a scholastic in the sense he reconciled human reason with Christian faith. Scholasticism was actually a popular movement in the twelfth and thirteenth centuries, gaining most of its momentum before and after the Crusades and bringing the reading of Greek, and the study of Aristotelian reasoning into Western Europe. The classics of Greece and Rome were never fully lost: from 476 AD onward, there was a continuous reading and study of the great Latin and Greek authors. The claim that the classical heritage was lost to Europe during the "dark ages" is both false and widely-accepted. Famous scholastics from that era like Pierre Abelard, Anselm, and Thomas Aquinas owe a great deal to Augustine. Augustine understood that Christianity was the same kind of truth that Plato and Aristotle discussed. That being the case, then all Christian concepts could be understood using reason, with the exception of God. To Augustine, God was beyond reason, and works in ways humans cannot understand. By showing that reason is a basis to understanding Christianity, Roman scholars could identify with the religion in a deeper manner. It made sense. He said, in one of his sermons, that “If you cannot understand, believe so you can understand.” In essence, he’s saying that faith is a precursor to knowledge. They are not contradictory. After all, you can't know anything unless you believe it. In fact, he believed that faith cleared the mind of confusion. “The skeptic concentrates on the weak points in human knowledge. The man of faith looks and see that there are points of strength also.” In his book City of God, he talks about how reason is a clear characteristic of God’s city. By espousing scholastic ideas, he made Christianity appealing to an even wider group of skeptics. However, this viewpoint of faith and reason will be subject to various interpretations during the middle ages: the Scholastics will present competing theories to explain how faith and reason work together.
In any case, Augustine, although he probably shouldn't be labeled a Scholastic in the technical sense of the word, clearly contains the main currents of Scholastic thought, if in embryonic form.