How does getting saved differ in the Old and New Testament periods?
How one gets saved does not differ in any period.
For by grace you have been saved through faith. And this is not your own doing; it is the gift of God, not a result of works, so that no one may boast. (Ephesians 2:8-9 ESV)
If you have read and understood the previous parts of this 'Salvation 101' series, you might be at the point where the Q&A here is obvious. Of course, how one gets saved is the same because it's about trusting God for His gift for the salvation of your soul.
It's not about behavior modification, an emotional or physical response, nor any need for additional sacrifices. Salvation can and does only come from God as a gift. It only happens that after Jesus was born that we could know His name. (PS: 'Christ' is His title, not His last name.)
All throughout the Psalms, for instance, King David trusted and exclaimed that salvation only comes from the Lord.
Salvation belongs to the LORD; your blessing be on your people! Selah (Psalms 3:8 ESV)
My soul longs for your salvation; I hope in your word. (Psalms 119:81 ESV)
But now the righteousness of God has been manifested apart from the law, although the Law and the Prophets bear witness to it-- (Romans 3:21 ESV)
For what does the Scripture say? "Abraham believed God, and it was counted to him as righteousness." (Romans 4:3 ESV)
This 'Scripture say' verse is quoted from...
And he brought him outside and said, "Look toward heaven, and number the stars, if you are able to number them." Then he said to him, "So shall your offspring be." And he believed the LORD, and he counted it to him as righteousness. (Genesis 15:5-6 ESV)
Final Note: There is a tremendous difference between salvation in the church-age and salvation in other times. But how one gets saved and what it means to be saved is absolutely the same - by grace through faith, for the righteousness that comes from God.
The difference is in the church itself, by which I mean the body of Christ, not just "a called out assembly." God has done something extraordinary in the church-age, which will the topic of future posts in another series, plus sprinkled messages here and there.