It seems you have a few questions here. I'll break it down.
1. How did people have different skin colors? There is no biblical, clear-cut answer, but GotQuestions has done their best to answer your question. Read their answer HERE.
2. What happened to people in Asia, and Africa during Jesus' time? How would they ever have the chance to hear about the Word?
First, it's important to understand that all people - no matter when/where they were born - are accountable to God whether or not they have heard the gospel. God has clearly revealed Himself in nature (Romans 1:20), in the conscience (Romans 2:15) and in the hearts of people (Ecclesiastes 3:11), so men are “without excuses” (Romans 1:20). A person does not end up eternally separated from God because of what they have not heard; he gets there because of his failure to respond properly to what God has revealed to Him. Deuteronomy 4:29 tell us that if a person genuinely seeks after God, He will make himself known. It is, therefore, the responsibility of a Christian to evangelize as many people as we can. We are not called to speculate who is saved, but to bring the good news to the nations (Matthew 28:19-20; Acts 1:8; Romans 10:15).
That being said, in Jesus' day (and even before his time), God did provide a way for people to be saved. It was through faith in God. Genesis 15:6 tells us Abraham believed God and that was sufficient for God to credit it to him for righteousness (see Romans 4:3-8). Listen to what GotQuestions writes:
What has changed through the ages is the content of a believer's faith. God's requirement of what must be believed is based on the amount of revelation He has given mankind up to that time. This is called progressive revelation. Adam believed the promise God gave in Genesis 3:15 that the Seed of the woman would conquer Satan. Adam believed Him, demonstrated by the name he gave Eve (v. 20) and the Lord indicated His acceptance immediately by covering them with coats of skin (v. 21). At that point that is all Adam knew, but he believed it.Summarily, those in Jesus' day - including inhabitants of Asia & Africa - were saved if they believed in God (and not in themselves or a false idol to save them from the sins). Today, God has revealed Jesus Christ to us, and it is through him alone that salvation is granted. Thus, it is our responsibility and calling - as Christ-followers - to tell people who have never of him the good news of his salvation!
Abraham believed God according to the promises and new revelation God gave him in Genesis 12 and 15. Prior to Moses, no Scripture was written, but mankind was responsible for what God had revealed. Throughout the Old Testament, believers came to salvation because they believed that God would someday take care of their sin problem. Today, we look back, believing that He has already taken care of our sins on the cross (John 3:16; Hebrews 9:28).
[Answered by Pastor HM]