Would you travel back to 33BC to see if it was all true about the son of god being born. and if it was true how would this change you?
You'd have to hang around for a while, since the usual estimates for the year he was claimed to have been born in are between 6 BC and 2 BC.
It would be useless as an answer to the question anyway, since it's a matter of faith. Finding out when he was born (assuming he was) is irrelevant to whether or not you believe he was the son of a god. You'd see a human baby. What you believed is a different thing entirely. I think it's probably true that a person probably called Jeshua probably preached a variant of Judaism that became Christianity. That doesn't mean I believe it's the true religion and all that goes with that belief. If I knew for a fact that he was born, it would change nothing. Why would it? At most, I might be able to find out what he actually said (as opposed to what he's now claimed to have said, which might or might not be the same thing) and I might be able to form an opinion as to whether or not he believed it himself.
Why do people assume that if you were to go back in time the lottery numbers would be the same? [..]
Because the hypothetical situation is that you have travelled back in time, not that you have travelled to a different reality. As a result, the common assumption is that the only changes are the ones that result from your actions. If that's not the case, then there's no point to this hypothetical discussion because it becomes "If you could travel to a different reality with no idea what it would be like, which one would you chose?", which is a completely meaningless question because choice is an irrelevant concept in that scenario. If you can't have any idea what you're going to, you can't make a choice.
For a one-way trip, I'd do the recent lottery thing too. I like reading about history, but I certainly wouldn't want to live in it. So I'd want to live here and now, but be rich. I'd also want to minimise unexpected changes resulting from my time travel.
There are other issues, though. If you go back in time within your own lifetime, there will be two of you. That could cause problems. Would the current you remain missing? Presumably - if you've gone to a different time, you've gone.