Another way of phrasing your question is whether Jesus Christ, born of the Virgin Mary, came before the "cave men" or vice versa. The cave man theory is not biblical at all. Evolutionists, who do not believe in Creation, theorize that humans came from apes that evolved over millions of years to produce a primitive man.
Man's creation, from the Biblical account in Genesis, simply excludes the evolution theory with all its beliefs, including the supposed existence of primitive cave men. We know that Jesus lived on this earth around two thousand years ago. Adam was created nearly six thousand years ago. Christ had to exist far sooner than any human, or indeed any life in the entire universe, since the Bible clearly states he created all things!
The apostle John states the following.
All things came into being through Him (Jesus), and not even one thing that was created came into being without Him (John 1:3, HBFV).
Neanderthals
Actually, what we refer to as cave men were actually fully human Neanderthals. It has been erroneously assumed that Neanderthals lived in caves because, to put it bluntly, they were too stupid to build a proper dwelling. Recent evidence, however, has shown that some Neanderthals did build places in which to live. Living in caves served a dual purpose, as they were many times used as burial grounds.
Neanderthal man is a direct descendant of Adam. He was therefore closer to the creation than we are. He was not a product of evolution and did not come from apes who, after who knows how many years, changed in primitive man.
The Creator creates
We see, in the second chapter of Genesis, Adam and Eve placed in the Garden of Eden. It was an ideal living condition for the first couple of humans to live. It was actually an idyllic living, with God instructing them in the way to live and in His laws. Scripture reveals that Jesus did exist before any cave men. In fact, He was the Creator who made the first humans!
Jesus said to them, 'Truly, truly I say to you, before Abraham was born, I AM' (John 8:58).
. . . and to make all see what is the fellowship of the mystery, which from the beginning of the ages has been hidden in God who created ALL THINGS through Jesus Christ (Ephesians 3:9).
Still, to us there is one God the Father, from Whom are all things, and we are in Him; and one Lord Jesus Christ, by Whom are all things, and we are by Him (1Corinthians 8:6).
Yes, Jesus Christ existed before any believed "cave men" took his first breath because everything we see was made by and for him.