Metro has several routes with frequent connections to Downtown Seattle. Use Metro's Trip Planner to plan a trip to CenturyLink Field from anywhere in the region. Just enter your trip's origin and your travel time and you'll receive a detailed itinerary telling you what mode of travel to take, where to catch it and how much it will cost.
There are not, at least ones provided by the stadium or club. Your best bet is the 545 or 550.
Not sure about shuttles, but definitely on public transportation and I think the light rail too.
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