The question about where the souls of Adam and Eve went after death, is the same question that applies to all pre-Jesus people. Obviously, the fate of unbelievers would be the same. But what about the Godly? There are two basic opinions about this.
1) Jesus’s death is sufficient for all, past, present, and future. The redemption of man was planned from the very beginning, when God told the serpent that Eve’s offspring would crush his head (Gen. 3:15). Further foreshadowing of redemption is seen in God’s covenant (Gen. 15) with Abraham and God providing the ram in place of Isaac (Gen. 22:13). Prophesies of redemption litter the Old Testament and is in fact the whole point of the Old Testament. Which is why Jesus said he did not come to abolish the law but to fulfill it (Matt. 5:17). Since redemption was always planned and since God is not bound by time, then the actual placement of Jesus’s death in time is irrelevant. Salvation was planned, therefore salvation was already done in the eyes of God.
That means the Old Testament Godly would be saved through Jesus, just like we are. They would go to Heaven.
2) The second theory takes a more linear view of God’s timing. Salvation could not come to the Old Testament Godly until after Jesus. Their souls would go to a resting place that existed prior to Jesus’s coming… Hades, Paradise, or Purgatory if you will. There is some Biblical evidence that could support this, such as Samuel coming “out of the ground” (1 Sam. 28:11-14) and Jesus telling the thief “today you will be with me in Paradise” (Lk. 23:43) when Jesus did not ascend to Heaven for another forty days (Acts 1:3). But it is important to note that even if a place like this did exist before, there is absolutely no support of it existing now. Once salvation was provided for the Godly, its inhabitants entered Heaven and this place was no longer needed.
The question now becomes, were Adam and Eve counted among the Godly? Did they repent and reconcile themselves to God? I think Genesis chapter four is sufficient to demonstrate that Adam and Eve continued to follow God and to train their children to follow God. They were a family that talked with and interacted with God, even though Cain made some bad choices. And so Adam and Eve should be counted as Godly.
Either option you choose from the two above, Adam and Eve would now be in Heaven.