How much do I know about the "Oral Torah"

Plenty. There was a great flowering of the Oral Law tradition during and after the Hasmonean period, and all of that would have been available to you at the end of the last century BCE.

Because you are educated but you're not Hellenized, your education wasn't an "academic" one of rhetoric, logic, aesthetics and physical exercise; it was a Jewish education, under a rabbi-teacher, in Bible, legends, law (halakhah), tradition and prayer. You would have learned to read the written Torah books (all the books of the Tanakh were written by then and Torah and Nevi'im -- the Prophets -- were canonized) and known the commentary on each verse. And you would have memorized a collection of laws that your teacher taught orally (having received it from his teacher) -- not the Mishnah (that of Rabbi Judah ha-Nasi, c. 200 CE), but an early mishnah (mishnah rishonah) from an earlier scholar; these are the ones Rabbi Judah utilized as his sources. If your teacher was very old school, he might have taught you a midrash halakhah ("legal midrash") to memorize -- a collection of the oral laws (halachot) recited in the order of the biblical verses from which they are derived (early versions of Mekhilta, Sifre, Sifra).

And of course you, your family and your neighbors would have been living the Oral Law in your daily lives, determining what you ate, when you prayed, how you observed your holidays, who you could marry, and so forth. You would have observed both laws and customs (minhagim) -- the customs might vary by region. When someone got married or divorced, it would all have been handled in accordance with the Oral Law, supervised if necessary by a rabbinical court of 3 (beth din). You knew what month it was and what day it was by the calendar fixed by the Oral Law. You knew for whom to mourn and how long to mourn according to the Oral Law. And when someone got involved in a dispute with a fellow, perhaps in business, or because of an unpaid loan, or perhaps there was a fight and someone was injured, or a fire and someone demanded compensation, you would have gone to a local rabbinical court and the outcome would have been determined by application of the Oral Laws. Developments like these would have been known locally and you would have been discussing them endlessly with your colleagues and neighbors -- they were the news of the day. And every time you heard a talk, lecture or sermon by a rabbi, he would have been delivering parts of the Oral tradition -- whether law (halakhah) or folklore, legends (aggadah).

So as I mentioned earlier, as a Jewishly educated individual at the end of the last century BCE, you would have known plenty about the Oral Law -- as it existed up to that time -- both from your educational background and your daily life, and it would have influenced your life in every conceivable way.