Thank you for the biblical question, “Is the Gospel found in the Old Testament?”
The Apostle Paul states the Gospel in I Corinthians 15, “Now, brothers and sisters, I want to remind you of the gospel I preached to you, which you received and on which you have taken your stand. 2 By this gospel you are saved, if you hold firmly to the word I preached to you. Otherwise, you have believed in vain.
3 For what I received I passed on to you as of first importance[a]: that Christ died for our sins according to the Scriptures, 4 that he was buried, that he was raised on the third day according to the Scriptures, 5 and that he appeared to Cephas,[b] and then to the Twelve. 6 After that, he appeared to more than five hundred of the brothers and sisters at the same time, most of whom are still living, though some have fallen asleep. 7 Then he appeared to James, then to all the apostles, 8 and last of all he appeared to me also, as to one abnormally born” (NIV).
In the above passage, the Apostle Paul states twice according to the Scriptures. The first citation is that Christ died for our sins and the second citation is that he was raised on the third day according to the Scriptures.
As one reads the Old Testaments, the words recorded in the Scriptures hinted at one who would remove sin forever. The Apostle Paul as a Pharisee knew especially the Scriptures very well.
In Acts 28, he says this, “ They replied, “We have not received any letters from Judea concerning you, and none of our people who have come from there has reported or said anything bad about you. 22 But we want to hear what your views are, for we know that people everywhere are talking against this sect.” 23 They arranged to meet Paul on a certain day, and came in even larger numbers to the place where he was staying. He witnessed to them from morning till evening, explaining about the kingdom of God, and from the Law of Moses and from the Prophets he tried to persuade them about Jesus. 24 Some were convinced by what he said, but others would not believe. 25 They disagreed among themselves and began to leave after Paul had made this final statement: “The Holy Spirit spoke the truth to your ancestors when he said through Isaiah the prophet:…” (NIV).
When the Holy Spirit sent Philip to meet an Ethiopian Eunuch, he was reading Isaiah 53. It was here that Philip shared the Good News about Jesus (Acts 8:30–35). Thus one can conclude that Isaiah 53 was a primary passage the early followers of Jesus used to attempt to persuade that Jesus fulfilled Isaiah 53.
Isaiah 53 says this, “Surely he took up our pain and bore our suffering, yet we considered him punished by God, stricken by him, and afflicted. 5 But he was pierced for our transgressions, he was crushed for our iniquities; the punishment that brought us peace was on him, and by his wounds we are healed. 6 We all, like sheep, have gone astray, each of us has turned to our own way; and the LORD has laid on him the iniquity of us all. 7 He was oppressed and afflicted, yet he did not open his mouth; he was led like a lamb to the slaughter, and as a sheep before its shearers is silent, so he did not open his mouth. 8 By oppression[a] and judgment he was taken away. Yet who of his generation protested? For he was cut off from the land of the living; for the transgression of my people he was punished.[b] 9 He was assigned a grave with the wicked, and with the rich in his death, though he had done no violence, nor was any deceit in his mouth.
10 Yet it was the LORD’s will to crush him and cause him to suffer, and though the LORD makes[c] his life an offering for sin, he will see his offspring and prolong his days, and the will of the Lord will prosper in his hand. 11 After he has suffered, he will see the light of life[d] and be satisfied[e]; by his knowledge[f] my righteous servant will justify many, and he will bear their iniquities. 12 Therefore I will give him a portion among the great,[g] and he will divide the spoils with the strong,[h] because he poured out his life unto death, and was numbered with the transgressors. For he bore the sin of many, and made intercession for the transgressors” (NIV).
In Isaiah there is the one who suffers, bears sins and dies and come back to life otherwise he wouldn’t be able to divide the spoils. There is the implication of resurrection. Jesus cited that as Jonah was three days in the belly of an sea creature so the Son of Man will be three days in the earth.
Did the Apostle Paul cite Psalm 22? We don’t know but it appears likely that he would have used that passage. That’s just my opinion.
As Paul reflected on the sacrifices at the Tabernacle, did he ponder whether there will be something or someone who would remove sin forever? I like to think so. The writer of Hebrews affirms that Jesus is the fulfillment of the Tabernacle.
As in the days of the Apostle Paul to this day, readers of the Old Testament and the New Testament differ in opinion whether Jesus fulfilled the Old Testament concerning the Suffering Messiah.
SUMMARY: Isaiah 53 is the clearest passage in the Old Testament that contains the forgiveness of sins for the people of Israel and for those who believe in that person.
For additional perspectives and discussion:
https://www.quora.com/Is-the-Gospel-found-in-the-Old-Testament