You make a couple of mistakes. First you wrongly assume that the Bible tells us that Adam was the first man who ever lived. It does not. Second, you misunderstand why Christ died. He died to pay for our sins, not to pay for what Adam did.
You are right in saying that the Adam of Genesis lived only about 6,000 years ago and in saying that people just like us have lived on earth for about 200,000 years.
To begin with, Bible chronology clearly indicates that only 4,000 years passed between the creation of Adam and the birth of Christ 2,000 years ago. But paleontologists, anthropologists and archcheologists all assure us that mankind has lived on earth far longer than 6,000 years. I believe this seeming conflict between Bible chronology and well established human history is easily resolved by understanding that the Bible does not tell us that Adam was, in an absolute chronological sense, "the first man". Rather, Adam and Eve were created by God and placed in Eden, a garden in the middle of an already widely populated world, to illustrate some very important lessons. Among them, the fact that none of us is deserving of eternal life. (By the way This understanding also answers the often asked questions, "Where did Cain get his wife?" and "Who were the people Cain was afraid might kill him?")
The only place in Scripture Adam is referred to as the "first" man is in 1 Cor.15:45-47. There Adam is called "the first man". But there we also find that Jesus is called "the second man". The context shows that the writer of those words was referring to Adam as the "first" man only in his relative chronological position to Christ. In other words, since Adam came "first" and Christ came "second", Adam came before Christ.
Some may object to this understanding of scripture, pointing out that their Bible tells them that God "made from one man every nation of men."(Acts 17:26) However, the words "one man" do not appear in Acts 17:26 in any ancient Bible manuscript. Most ancient manuscripts simply say that God "made from one every nation of Men." Other ancient Greek manuscripts, from which this portion of the Bible is translated, tell us that God "made from one blood every nation of men." For this reason The Amplified Bible here reads God "made from one [common origin, one source, one blood] all nations of men." The New English Bible translates this verse to tell us that God created every race of men from "one stock." So, Acts 17:26 can only be used to confirm that all people on earth are descended from ancestors who came from the same gene pool, and that all people on earth have the same original source of origin, a teaching which fully agrees with the findings of modern science. This verse does not say, in any Greek manuscript, that mankind's common origin was one man.
Most other objections to this understanding of Scripture come from those who adhere to the doctrine of "The Fall" of mankind. This doctrine is based on what I believe is a misunderstanding of the apostle Paul's words in Romans 5:12-20 and 1Corinthians 15:21,22.
The Bible clearly tells us that God will hold each one of us responsible for his or her own unrighteousness, not for Adam's. (Romans 14:10-12, 2 Corinthians 5:10) And the Scriptures say that we all need the forgiveness God offers us through Jesus Christ, because we have all personally "sinned" and have all personally "fallen short of the glory of God." (Romans 3:23)
I do not believe the Bible teaches that mankind "fell". Rather, I believe it tells us that God originally created the human race as free people. Free to do both right and wrong. In the exact same way we are free to do so today. Unfortunately we often choose to do what is wrong rather than what is right. God, however, cannot do wrong. For God is "incorruptable".(Romans 1:23) So, because we can and often do behave unrighteously, and because God cannot and never does behave unrighteously, we are less righteous than God. And, because "all unrighteousness is sin" we are all born "sinful".(1John 5:17, New American Standard Bible; Psalms 51:5)
Being able to do wrong, Adam was, from his very beginning, also less righteous than God. And he later proved his "sinful" condition by his behavior. Because Adam in paradise could not manage to obey one simple command from God, he clearly demonstrated that he and the entire human race, including those who had lived before him and those who would live after him, were far less righteous than God.
So, with these things in mind, Paul accurately referred to Adam when he wrote,"by one man's disobedience many were constituted sinners." (Romans 5:19, Amplified Bible) This is true because Adam's disobedience demonstrated that the entire human race was not only capable of doing wrong but incapable of not doing wrong. So, after Adam failed a simple God given test of his righteousness, God had good reason to retroactively condemn the entire human race as being deserving of the deaths they had been suffering, and undeserving of eternal life, a gift God had not yet given to any human being.
I believe those who adhere to the doctrine of "The Fall" also basically misunderstand the events which transpired in Eden.
The Genesis account clearly indicates that Adam and Eve were created mortal with a dying nature just like us. The story of Adam and Eve told in Genesis makes clear that their being able to live forever was not a part of their original physical nature. Rather, Adam and Eve's ability to live forever depended entirely on their eating from a tree "in the middle of the garden" of Eden, "the tree of life".(Genesis 2:9) Genesis tells us that Adam and Eve were going to be allowed to continue to eat from that tree only if they passed a God given test, a test which we are told they failed. After failing that test God expelled Adam and his wife from the Garden of Eden and prevented them from ever again eating from "the tree of life".
Genesis indicates that had Adam and Eve been allowed to continue eating from "the tree of life" their lives would have been prolonged indefinately.(Genesis 3:22-24) But when God prevented them from ever again eating from "the tree of life" they died what were apparently natural deaths. A careful reading of the Genesis account shows us that living forever would have been as unnatural for Adam and Eve as it would now be for us.
Genesis does not indicate that Adam and Eve originally had eternal life programmed into their genetic codes by God and later had their genetic codes reprogrammed by God in order to remove eternal life from those codes. Rather, Genesis indicates that Adam and Eve would have lived forever only if God had graciously given them eternal life from an outside source, "the tree of life".
Of course, that "tree of life" was meant to picture Jesus Christ.
For, as we have seen, God was going to give Adam and Eve eternal life from an outside source, "the tree of life", only if they passed a very simple test. And the Bible tells us that we will be given eternal life from an outside source, Jesus Christ, only if we pass a very simple test. That test is to simply believe in our hearts that Christ's death was sufficient payment to buy every human being God's full forgiveness, forgiveness for both our sinful nature and our sinful acts.
By the way, this understanding of scripture is not by any means new, it has just not yet been widely embraced. The Roman emperor Julian the Apostate (A.D. 331-363) held this understanding of scripture, but he thought erroneously it could be used as a counterpoint to Christianity to restore paganism. Isaac de la Peyrere, a Catholic priest, held this understanding of Scripture in 1656. For his efforts he was forced to recant and his books were burned. In 1860, one year after Charles Darwin published The Origin of Species, Bible scholar Edward William Lane published this understanding, but anonymously to escape reprisals. Today this understanding is being advanced by Christians such as Richard Fischer. Fischer graduated from the University of Missouri with a Bachelor of Science degree. His first article on religion was published in The Washington Post in 1986. He received his master's degree in theology in 1992. He has published articles in Perspectives on Science and Christian Faith, and has reviewed articles for publication in Christian Scholar's Review. He is a member of American Scientific Affiliation, Interdisciplinary Biblical Research Institute, Evangelical Theological Society, and he is listed in Who's Who in Theology and Science.
Fisher's book on the subject is entitled The Origins Solution. Some sample chapters of it can be read and copies of it can be ordered from his publisher's web site. http://www.orisol.com/
