"Typology" refers to the habit of reading the Old Testament as foreshadowing the New Testament. Thus, Adam in the Old Testament foreshadows Christ in the new; the Tree in the Old Testament foreshadows the Cross in the New; Mary in the Old Testament foreshadows Eve in the New; the story of Abraham and Issac in the Old Testament foreshadows the actual crucifixion in the New, and so on.
The passage from Augustine to which you refer has been interpreted in similar ways. Here is one example:
See also this link: