If one was travelling back to Israel at the time of Christ, it seems to me that it may not be all that easy to identify Him until he started to get a reputation as a preacher*:
Many people shared the name. Christ's given name, commonly Romanized as Yeshua, was quite common in first-century Galilee. (Jesus comes from the transliteration of Yeshua into Greek and then English.) Archaeologists have unearthed the tombs of 71 Yeshuas from the period of Jesus' death. The name also appears 30 times in the Old Testament in reference to four separate characters—including a descendent of Aaron who helped to distribute offerings of grain (2 Chronicles 31:15) and a man who accompanied former captives of Nebuchadnezzar back to Jerusalem (Ezra 2:2)....Now, I don't know how big a town Nazareth was at the time, but it seems a very good bet that there was more than one "Jesus of Nazareth", and (I would guess) another local "Jesus, son of Joseph" sometime before our Jesus hit 30.
What was Jesus' last name? It wasn't Christ. Contemporaries would have called him Yeshua Bar Yehosef or Yeshua Nasraya. (That's "Jesus, son of Joseph" or "Jesus of Nazareth.") Galileans distinguished themselves from others with the same first name by adding either "son of" and their father's name, or their birthplace.
And while on the topic: I see that Wikipedia has at this entry (under the subheading "Jesus") a list of science fiction books and stories featuring time travel back to Jesus' time.
The one story idea which I don't see mentioned there is a time travelling expedition from the future to make sure the Jesus story as shown in the Gospel happens. (Sponsored by someone from the Catholic Church who has lost faith, but figures on utilitarian grounds that the net benefits of belief to society would be worth the fraud, and employing one of the modern illusionists of the kind we see on TV now apparently performing convincing tricks in the middle of the street.)
Having thought of this idea quite some time ago, it has had the unfortunate effect that when hearing a Gospel reading at Church, my mind often wanders to how a modern illusionist would replicate the effect. Certainly, the glowing or shining appearance of angels, especially at night, is easy to imagine with simple UV light; unconsumed burning bushes (yes, I know, wrong Testament) would be easy as, too, and so on.
* Yes, I know a bit of Jesus identity confusion did feature in one episode of Red Dwarf, and I'm not being completely original.