How to prove Christianity for those who never heard of it?
You can't use the Bible until you prove the Bible. You can't invoke common sense because the gospel seems like "foolishness". (1 Corinthians 1:18) (1 Corinthians 1:23) Archaeology doesn't work because mainstream archeology doesn't agree with Christian archaeology. Ditto for science. You can't use the subjective argument that once you believe "it just feels true".
It seems you're saying that common sense, archaeology, and science all debunk the Bible. I appreciate the honesty.
However, it does all ultimately balance upon the resurrection of Jesus Christ.
If it can be definitively proven that he did not rise from the dead, then Christianity would be dead even if we found hundreds of legitimately fulfilled prophecies in the Bible.
Conversely, if it can be definitively proven that he did rise from the dead, then all the problems in the Bible would really be moot.
Why die for a lie?
This is the go-to argument of apologists. Islamic suicide bombers, for instance, are indeed willing to die for their beliefs but they are not in a position to know with certainty whether or not their beliefs are actually true or false. However, the eleven disciples who supposedly saw the resurrection were in a position to know whether or not their beliefs were true or false. And it is emphasized that a hallucination cannot explain their genuine beliefs in the resurrection because hallucinations are not shared experiences.
However, this argument is seriously disingenuous. The earliest known Christian who refused the opportunity to recant and go free was Polycarp. Polycarp was born after Christ had died, so he could not have been a witness to the resurrection. To our knowledge, none of the eleven disciples who were present were given the same opportunity.
The canonized version of the Book of Acts says that James was slain by the sword at the behest of a king. It does not say that he was given the opportunity to recant. We have no idea what happened. It's not reasonable to assume that a discussion even took place at all.
Peter, even by secular accounts, was executed. However, we don't know the details of that, either. It's possible that his only choices were to either admit to the "crime" of being a Christian and receive a swift death, or else suffer torture until he confessed. Christians were scapegoated for the great fire of Rome, and it's plausible that Peter was executed because he represented the Christians. There is no reason to assume that he was given the same opportunity as Polycarp because Polycarp came a hundred years later, which means that there was another emperor in place.
Paul and Stephen did not witness the resurrection, and all mentions of them in the "Why die for a lie?" argument are totally dishonest. Jesus Christ rose from the dead and stayed with the disciples for around a month, and then left earth. He has never physically returned. This occurred long before Paul and Stephen had their experiences. Furthermore, recall that when Paul and Stephen had their experiences, there were other witnesses present who did not experience the same thing. So their experiences were *effectively* hallucinations, even if Jesus did appear to them in some "real" way.
Even taking the Bible and all non-canonical Christian works as 100% true, there is no known person in all of history who saw Jesus Christ rise from the dead and then later refused to recant their testimony under duress. The case for the resurrection absolutely cannot be made unless you relax the burden of proof to the point that we can prove the existence of things like Big Foot, alien abductions, and the Loch Ness Monster.