How Did God Save During the Old Testament Period?
The New Testament declares that the gospel was preached to Abraham beforehand (Galatians 3:8). And Jesus said of this ‘Your father Abraham rejoiced to see my day: and he saw it, and was glad.’ (John 8:56) How were the people of the OT saved? By works? By Animal sacrifice? (Heb 10:4) No. They were saved by Christ and Christ alone (see Heb 9:15). The gospel they knew was a shadow but they were still saved by the blood of Christ. There is no other way of salvation except by "the lamb who was slain from the foundation of the world." (Revelation 13:8)
Submission to the Levitical sacrifice under the old administration was a yielding of God's way of salvation in the OT. The regenerate Jews realized that these were but types of the reality that was to come. The fact that the Jews apprehension of this was incomplete is irrelevant. As for people such as Cornelius and Lydia who are said to be God fearing, worshipers of God, why was the gospel preached to them if they were already saved? The good news was preached to those in the transition period (the book of Acts) so that those who already knew Christ in shadows would now know the fulness of revelation. (See Romans 4:17–24 & Galatians 3:6–29) The only alternative would be to keep the fulness of the good news from them, which would be problematic. Enlightened saints in the transition period were waiting for the Christ and wanted to behold him (Luke 2:26).