It does, at least in type, if we know where to look. In Galatians 3:13, Paul said that Christ was made a curse for us: for it is written, Cursed is every one that hangs on a tree, and that was written by Moses back in Deuteronomy 21:23. We know that the tree upon which Christ hung was the cross Acts 5:30, Acts 10:39, Acts 13:29, 1 Peter 2:24. As you said, Jesus quoted from Psalm 22:1 while he was hanging on the cross. In that psalm, we see descriptive language that speaks of death by crucifixion, like Jesus' bones being out of joint, and his hands and feet being pierced. In John 3:14, Jesus told Nicodemus that he must be lifted up as Moses lifted up the serpent on the pole in the wilderness in Numbers 21:8, and his lifting up referred to his crucifixion on the cross. These are just some examples where Christ's crucifixion was foreshadowed in the Old Testament.