Why can't I kill Alduin during the first fight?
Solution 1:
There was aparently a bug in early-versions of the game, potentially still existing here today, where he is not able to take damage.
Some had luck in restoring his ability to be damaged by exiting and reloading the most recent save.
Since you're playing the Xbox version, the best bet would seem to be to reload the most recent save. There doesn't seem to be any definitive fixes among the results I've found.