In 1 Cor, when Paul is laying out the the tradition of the resurrection to his readers, he lays out a tradition with his own testimony at the end.
1Cr 15:3-8 For I delivered to you as of first importance what I also received, that Christ died for our sins according to the Scriptures, and that He was buried, and that He was raised on the third day according to the Scriptures, and that He appeared to Cephas, then to the twelve. After that He appeared to more than five hundred brethren at one time, most of whom remain until now, but some have fallen asleep; then He appeared to James, then to all the apostles; and last of all, as to one untimely born, He appeared to me also. [NASU]
However, there is no quotation in Old Testament that the Messiah would be raised on the third day. This, then, must be an example of midrash, something applied to the Messiah that on the surface speaks of something else. For example, Matthew (2:15) applies a statement in Hosea (11:1) to Jesus, "Out of Egypt I have called my son."
What Old Testament Scripture is Paul drashing* from here?
*An Anglicization of the Hebrew verb, drash meaning "seek" and used by the rabbi to explore Scripture and find it's application to other issues which were not stated in the text.
Why Assume Paul Means the Old Testament
I conclude that he is referring to the Old Testament and not a Christian document because Paul uses the word graphe 14 times in his letters. 8 of these are followed by direct quotations of the Old Testament.
Rom 4:3 For what does the Scripture say? "ABRAHAM BELIEVED GOD, AND IT WAS CREDITED TO HIM AS RIGHTEOUSNESS."
Rom 9:17 For the Scripture says to Pharaoh, "FOR THIS VERY PURPOSE I RAISED YOU UP, TO DEMONSTRATE MY POWER IN YOU, AND THAT MY NAME MIGHT BE PROCLAIMED THROUGHOUT THE WHOLE EARTH."
Rom 10:11 For the Scripture says, "WHOEVER BELIEVES IN HIM WILL NOT BE DISAPPOINTED."
Rom 11:2 God has not rejected His people whom He foreknew. Or do you not know what the Scripture says in the passage about Elijah, how he pleads with God against Israel? 3 "Lord, THEY HAVE KILLED YOUR PROPHETS, THEY HAVE TORN DOWN YOUR ALTARS, AND I ALONE AM LEFT, AND THEY ARE SEEKING MY LIFE."
Gal 3:8 The Scripture, foreseeing that God would justify the Gentiles by faith, preached the gospel beforehand to Abraham, saying, "ALL THE NATIONS WILL BE BLESSED IN YOU."
Gal 3:22 But the Scripture has shut up everyone under sin, so that the promise by faith in Jesus Christ might be given to those who believe. (Here it is used as a synonym for Law which appears in both 21 and 24.)
Gal 4:30 But what does the Scripture say? "CAST OUT THE BONDWOMAN AND HER SON, FOR THE SON OF THE BONDWOMAN SHALL NOT BE AN HEIR WITH THE SON OF THE FREE WOMAN."
1Ti 5:18 For the Scripture says, "YOU SHALL NOT MUZZLE THE OX WHILE HE IS THRESHING," and "The laborer is worthy of his wages."
In the others, he is clearly alluding to the Old Testament.
Rom 1:2 which He promised beforehand through His prophets in the holy Scriptures,
Rom 15:4 For whatever was written in earlier times was written for our instruction, so that through perseverance and the encouragement of the Scriptures we might have hope.
Rom 16:26 but now is manifested, and by the Scriptures of the prophets, according to the commandment of the eternal God, has been made known to all the nations, leading to obedience of faith;
2 Tim 3:15 refers to the Holy Writings which Timothy had been trained in before he heard Paul's preaching. As Timothy's mother and grandmother were Jewish, logically, those holy writings would be the Old Testament. Paul continues with:
2Ti 3:16 All Scripture is inspired by God and profitable for teaching, for reproof, for correction, for training in righteousness;
If we add in grapho, the verb form of graphe, Paul uses that 62 times. Sometimes he refers to his own letters or general letters. More often, he quotes from the Old Testament. For example:
Rom 15:9 and for the Gentiles to glorify God for His mercy; as it is written, "THEREFORE I WILL GIVE PRAISE TO YOU AMONG THE GENTILES, AND I WILL SING TO YOUR NAME."
1Cr 9:9 For it is written in the Law of Moses, "YOU SHALL NOT MUZZLE THE OX WHILE HE IS THRESHING." God is not concerned about oxen, is He?
2Cr 9:9 as it is written, "HE SCATTERED ABROAD, HE GAVE TO THE POOR, HIS RIGHTEOUSNESS ENDURES FOREVER."
If we add in the Paul narratives from Acts, there are 2 times when graphe is used in reference to Paul (Luke uses the word at other times, also in reference to the Old Testament). These are:
Act 17:2 And according to Paul's custom, he went to them, and for three Sabbaths reasoned with them from the Scriptures,
Act 18:28 for he powerfully refuted the Jews in public, demonstrating by the Scriptures that Jesus was the Christ.
In Acts 17, he is speaking in the synagogue. That would require basing his sermon off an Old Testament passage. Even if he then began quoting rabbinic interpretations of the Old Testament, he would have to start from an Old Testament scripture. The same is true of debating the Jews in public. He had to base his reasoning first on written Torah.
Which leaves only two occurrences of graphe, those under discussion:
1Cr 15:3 For I delivered to you as of first importance what I also received, that Christ died for our sins according to the Scriptures, 4 and that He was buried, and that He was raised on the third day according to the Scriptures,
From Paul's usage of the word graphe and grapho elsewhere overwhelmingly being to the Old Testament, we may reasonably conclude that in 1 Cor 15:3 and 4, he is also making his basis on the Old Testament (if you include the times that Paul alludes to the Old Testament or quotes it without an introduction, the evidence is even more overwhelming that he is thinking of the Old Testament).
Rabbinic Interpretations of a Dying Messiah
There are rabbinic and Essenic interpretations of Old Testament passages showing that the Messiah would die. Specifically, that the Messiah ben Yoseph or Messiah ben Levi would die for the sins of Israel. One of the Old Testament passages they cite as their basis is Isaiah 53. While modern interpretation sees this chapter as referring to Israel, the rabbinic commentaries until the 11th century understand Isaiah 53 as referring to the Messiah.
The Messiah---what is his name? Those of the house of Rabbi Yuda the saint say, the sick one, as it is said, 'Surely he had borne our sicknesses.' (BT Sanhedrin 98b)
And when Israel is sinful, the Messiah seeks for mercy upon them, as it is written, "By His stripes we were healed, and He carried the sins of many; and made intercession for the transgressors." (B'reshith Rabbah)
The Holy One gave Messiah the opportunity to save souls but to be severely chastised: and forthwith the Messiah accepted the chastisements of love, as is written, "He was oppressed, and he was afflicted." And when Israel is sinful, the Messiah seeks mercy upon them, as it is written, "By his stripes we were healed," and "He carried the sins of many and made intercession for the trangressors." (Bereshith Rabbah, Rabbi Moshe Hadershan)
Also interpreting Zechariah 7:10
"Messiah son of Joseph was slain, as it is written, "They shall look unto me whom they have pierced; and they shall mourn for him as one mourneth for his only son" (Suk. 52a)
To conclude that Paul is drawing from the same tradition in his interpretation of Scripture to say that the Christ would die for our sins is completely reasonable.