Why is the kanji for うん(運) the same as the kanji for 運ぶ? Did the kanji just somehow end up being the same, or were the two meanings related somehow? All I can think of is some kind of "carrying luck", but that doesn't seem right.
I think there are multiple interpretations of this character, but it's clearly a combination of 辶 (from 辶) and 軍, which suggests the movement-related meaning came first and "luck" was a derived meaning. But how was it derived? Here's what Henshall has to say:
辶 is movement 129. 軍 is army 466 q.v. Some scholars take the latter in a literal sense, giving army on the move and by association transportation and the fortunes of war. Others take it to act phonetically to express round, as well as lending its own connotations of both circle and vehicle (from a circle of vehicles), thus giving a meaning of vehicles rolling along, and hence transport. Luck is then felt to stem from an association between fortune and circular/cyclic movement.
It seems like the exact way it came to mean "luck" is unclear, but there you have a couple possible ways it could have happened.
This is speculation on my part, but I believe it is possible that when 運 is used to mean one's luck or fortune, it may be interpreted as "[that which is] carried".
To illustrate, consider the word 運命(うんめい), meaning destiny or fate. Taking the two kanji apart, we have:
運ぶ (はこぶ) meaning to carry
and 命(いのち) meaning life.
So, synthesized, it would mean "the life that one carries" or even "the life that one is made to carry".
As for why 運 without any object would refer to one's luck, it may refer to that which is naturally and constantly possessed and thus the "default in absence of others". A man stripped of all status and possessions would still have his life, as it is part of him. And yet it can be the whole of him too, since it seems that one's destiny is seen to be inescapable, decided by the divine and placed upon one's shoulders.