Phoenix Sky Harbor International Airport: This is the airport to use. For Old Town/downtown Scottsdale, take Uber/Lyft or a taxi: roughly 15–25 minutes without heavy traffic and usually about $25–45 before surge pricing. Public transit is the budget fallback, not the smart default with luggage: take the PHX Sky Train to 44th St/Washington and connect onward by Valley Metro bus; allow about 60–75 minutes and roughly $2–4.
Scottsdale Airport: A general-aviation and charter airport about 20 minutes north of downtown, with limited scheduled JSX service. Use a pre-arranged car, rental car, or rideshare; expect roughly 20–30 minutes to Old Town and fares that vary sharply by demand. It is convenient only if your flight actually lands here.
Phoenix–Mesa Gateway Airport: A secondary airport in southeast Mesa, mainly useful for the specific low-cost flight you found—not for Scottsdale access. Rent a car or use rideshare; allow about 40–60 minutes to Old Town in normal traffic and roughly $45–75 by rideshare. There is no quick rail link to Scottsdale.