I have heard people argue that people in the old testament had faith in that they would be saved through Jesus Christ but there is no mention of Jesus Christ in the old testament
In the Bible the key think is that person must be righteous. And if person is righteous, he will live.
These will go away into eternal punishment, but the righteous into eternal life.
Mat. 25:46
For the wages of sin is death, but the free gift of God is eternal life in Christ Jesus our Lord.
Romans 6:23
Righteous person lives, if he is faithful (loyal) to God.
Behold, his soul is puffed up. It is not upright in him, but the righteous will live by his faith.
Habakkuk 2:4
God could forgive their mistakes even without Jesus. But, I think those who died before Jesus were saved by this way:
Most assuredly, I tell you, the hour comes, and now is, when the dead will hear the Son of God's voice; and those who hear will live.
John 5:25
The tombs were opened, and many bodies of the saints who had fallen asleep were raised;
Matt. 27:52