Does the Millenium Hotel Have an Airport Shuttle?

Yes, they do.  Upon arrival at the Anchorage airport, claim your bags first, and then go to the hotel kiosk in the baggage claim area.  Find and push the Millennium button to contact them to have them pick you up.  Their shuttle is not a continuously-running one; it is on-demand only.  They won’t know to come get you unless you call them.  The hotel personnel will tell you where to go at the airport to be picked up.