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Wolverine's whole body has an Adamantium layer underneath his skin. Adamantium is an indestructible metal, as shown in X-Men: The Last Stand but in X-Men 2 a policeman shoots Wolverine in the face, outside Bobby's house. The bullet appears to pass through the Adamantium skeleton and kill Wolverine (at least for a couple of minutes).
How did the bullet pass through the Adamantium layer?
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.
Read the first couple paragraphs here.
TL;DR: The bullet did not penetrate the adamantium layer, he's just been hit in the skull by a lump of metal traveling at probably something over 1,000 feet per second, which is why he got knocked out which seems the most plausible explanation. The bullet was just embedded in the flesh that's around his skull.
he's just been hit in the skull by a lump of metal traveling at probably something over 1,000 feet per second - there's a good chance he got a nasty concussion, too.
The film's official novelisation confirms that the bullet, although penetrating the skin and breaking into pieces, didn't (and couldn't) penetrate his impenetrable adamantium skull. He's not dead, merely unconscious.
She yelled at him some more, partly to purge her own terror, but most of all to keep attention away from Logan. She knew the adamantium interlaced with his skeletal structure meant that his bones couldn’t be broken. All that bullet had likely done, aside from breaking the skin—which was decidedly messy—was give him a royal headache. More importantly, though, his healing factor would be speedily dealing with both the wound and the headache. She didn’t know what he could do once he recovered, but it would be one more asset than the kids had right now.
and
Logan’s eyes fluttered as the shattered remains of the officer’s bullet fell from the healing wound. Rogue was right; his head was murder. This was a great power, no argument there.
t his bones covered resp. didnt his bones react with the adamantium? The skull is a large bone more or less directly under the skin, that´s why in ths case it`s nearly the same, but the rest is another story.