Wolverine was shot in the head and he is knocked out, but to me it clearly looks like the bullet is sitting in his head in the video in your question. As his skin heals, the bullet pops out. (Valorum's answer confirms this with a quote from the novelization.)
So the bullet does not look like it passed through his Adamantium-bonded skull. And if it had, and his brain was damaged by the bullet, he would have suffered memory loss after the brain regenerated. This happened in X-Men Origins: Wolverine when Stryker shot him with two Adamantium bullets (or Vibranium, possibly). In X2, there is no memory loss because the bullet did not damage his brain.
Wolverine's Adamantium skull could have two bullet holes in it from Stryker's Adamantium bullets (see this answer), so he could perhaps get damaged by a regular bullet at the exact same spot now, if he was extremely unlucky. However, according to this answer the Adamantium plating on his skull could have regenerated as well. In any case, the regular bullet in X2 clearly did not hit the same spot. Compare the Adamantium bullet holes in the first picture with the regular bullet "hole" in the second:
Anyway, the answer to the title of your question is "no". Normal bullets do not go through Adamantium.