Nope. First of all there was no flood, Noah and the Ark is a story used to convey some wisdom or teaching. Secondly unicorns wouldn't appear naturally, a feature such as a horn would not be a viable step evolution simply because a horse wasn't made to ram to defend itself.
Is It True That Unicorns Existed Before Noah's Ark But He Could Only Find One So The Unicorns Went Extinct?
No this is a complete myth and if you read the old testament not all animals "went in two by two"
No, the King James Bible mistranslates the Hebrew word for a wild ox(most likely the auroch) as unicorn. The Bible never said anything about unicorns in the original Greek and Hebrew.