First of all, these subject is a lot easier to research if you know that the “mystery religions” being referred to here are the various incantations of Gnosticism.
Gnosticism was was not unique to that erra of history. Versions are documented long before the time of Christ and it can still be found in various flavors today. However it did perhaps reach its popular climax about the same time Christianity was spreading through the 1st century modern world.
The various ideas of Gnostic thought had a profound formative effect on early Christianity. However this effect was more reactive than assimilitive. The teachings of Christianity stand in stark contrast to all forms of Mysticism. Yet, as is ever the case, there are plenty of folks willing to promote various brands of syncretism. During the first couple of centuries, Gnostism and the various Gnostic cults that sprung up were the first major heresies Christianity had to deal with.
You will in fact find that much of the teaching of the early Christian apostles was specifically directed against Gnostisms. For example, the shorter letters written by the Apostle John were directed squarely against a prevalent Gnostic teaching that Jesus did not actually have a physical body but was just a vision, something like a hologram projected into this physical world. Contrary to modern perceptions, the first obstacle that Christianity had to overcome was not convincing people that Jesus was God but convincing people that God actually took on flesh and became a man.
The mystic religions of the time said that this was shameful, that a pure holy God would never defile himself by becoming fleshly. John explained the Christian doctrine to the contrary starting his letter with these words:
1 John 1:1-2 (ESV)
1 That which was from the beginning, which we have heard, which we have seen with our eyes, which we looked upon and have touched with our hands, concerning the word of life— 2 the life was made manifest, and we have seen it, and testify to it and proclaim to you the eternal life, which was with the Father and was made manifest to us—
In other words, Jesus was touchable. His body was real. He goes on to treat ideas such as “light” that were often used in mystical ways, showing that in Christianity their real meanings were quite concrete.
Early Christian history is littered with evidence of their opposition to mystic teachings. According to some popular mystic ideas, salvation was to be freed from this physical world: united with a spiritual dimension by discarding the shells that we are trapped in. Christianity stood in opposition to this, claiming salvation through the work of God redeeming both the spiritual and eventually the physical worlds.
If you follow the development of Christian creeds, you will find the points that the church felt they needed to emphasize and expand on in each successive expression of their core beliefs were usually reactions to some popular mystic idea having found its way into the practice of Christians of the time. You can find some examples of this progression in my answer to a question about the creeds here.