There is a bus that comes from the airport too downtown Spokane. It is number 60. Once in Spokane the #2, a shuttle bus , leaves Spokane and goes to the driveway of the Madison Inn. Unfortunately, that bus does not go to the front door of the Madison. Hope this helps you.
There is not unfortuantely but you can call orange cab taxi. I work front desk at the Madison and we use them constantly for guest recommendations.
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