Here is a little info for those who are new to Skyrim and who wants to try this like I did.

Do not try to kill Alduin early in the game, I tried hunting him down and killing him and I finally got hold of him one day near Rorikstead busy reviving another dragon , So I made my way up the hill as fast as I can using Whirlwind Sprint and all of my stamina. When I finally got to him , I pulled out my Bow and fired a shot at him. Guess what I got him , but he immediately ended the ritual and flew off into the sunset just to be lost to me again.

If you want to kill Alduin follow the Main Quest and you will have your day. Believe me the wait and the chase is long before you find him, and you will not be able to kill him.