This is a fascinating legend that I hadn't come across before: I haven't managed to trace its history back earlier than the late 1800s, the date of Sergey Vasilyevich Ivanov's painting in the Church of St Mary Magdalene in East Jerusalem. (Other icons showing the red egg are apparently all much more recent.) It's apparently a Russian Orthodox tradition, so I presume you'd need to be able to read Russian sources to examine its history with any thoroughness. If you have any older sources, it'd be interesting to take a look at them.

Anyway, your actual question is:

How likely, would have Tiberius known about Christianity ?

The Jesus movement during Jesus' own lifetime necessarily lacked most of the religious practices that would later be called 'Christianity' - such as the Eucharist, the gospels, veneration of the cross, and so on. We can't reliably date the practice of venerating Jesus as the Messiah/Christ to Jesus' lifetime, and unless you're a believer, we can't know how far back the belief in the resurrection goes earlier than Paul writing about it in 1 Corinthians 15 (50s CE?). It's arguable whether you should call it 'Christianity' before these things come along.

So the question would really be: is it possible that Tiberius would have heard of the Jesus movement in Syria/Judaea in the late 20s/early 30s CE?

To some extent it's speculative. Some news from the provinces certainly did penetrate to Rome, and there are inscriptions showing that the emperor's office was engaged with provincial affairs in some respects. For comparison, Pliny the Younger's letters to Trajan show him consulting the emperor about matters that in some ways look relatively trivial.

Then again, Tiberius left Rome around 26-27 CE and lived in semi-retirement on Capri for the rest of his life, leaving the management of Rome to Sejanus until 31 CE. Additionally, we have Josephus' report that when Pilate came to Rome to face discipline, Tiberius was gone: it seems unlikely that there was much communication going on. Modern biographers tend to suggest that from Sejanus' downfall in 31 until Tiberius' death in 37, the empire just kept ticking. Literary sources don't really support the idea that the (still semi-retired) emperor might have taken much interest in a local cult in one of the most distant provinces. When envoys did come to Tiberius about affairs in the far east, in Armenia, in 35 CE, he responded, but delegated the job: he doesn't seem to have been closely involved himself.

So I'm going to say: it's possible, but in the specific case of Tiberius, I think we've got reason to think it very unlikely.