I'm not sure if this is precisely what you mean, but the oldest rabbinic text (so old in fact that it's perhaps better described as proto-rabbinic) is Megillat Taanit. It comprises a list of dates on which it is forbidden to fast, or on which it is forbidden to recite a eulogy. There is an accompanying text that scholars term the scholion, which serves as a commentary on Megillat Taanit, describing what happened on each of those days. The text (including the scholion) is so old as to be referred to in the Mishna (eg: Taanit 2:8), and is our earliest source for the story of the miracle on Hanukkah.
If you are asking for information on the oldest text (ie: not which text was composed before all of the others, but which textual remains are, physically, of the greatest antiquity), then the answer would be the material found in the Dura Europos synagogue in Eastern Syria, dating from 244-256 CE. To quote from Stephen Fine, "a parchment fragment discovered [in Dura Europos] in 1932 is highly reminiscent of rabbinic prayer texts; inscriptions closely parallel Targumic paraphrases of Scripture in Aramaic; and the wall paintings suggest important parallels with midrashic sources." [source].
While not the oldest, strictly speaking, a very old (and very interesting) example of this phenomenon can be found in the synagogue in Rechov. Dating to the 5th-6th centuries, there is an inscription there that, in the words of Stephen Fine, is the earliest preserved halakhic text yet discovered. The text of the inscription directly parallels passages in Tosefta Shevi'it 4:8-11, Sifre Deuteronomy 51 and Jerusalem Talmud Demai 2:1, 22c-d and Shevi'it 6:1, 36c. According to Prof. Saul Lieberman, it was probably the transcription of a letter sent from a bet din to the synagogue in Rechov, treating of the laws of tithes in the sabbatical year. The full text can be read in English at the same site to which I linked above, which was Stephen Fine's article, "The Rehov Synagogue Inscriptions: The Earliest Preserved Text of the Talmudic Literature".