I don't even think it's an issue of getting off Amtrak in Seattle instead of Tukwila - the 2:24 PM arrival into Tukwila starts from Seattle-King Street, it is not a continuation of any cross-country route. Don't even bother boarding that train in the first place. The station in Tukwila is just a platform and a parking lot in the middle of nowhere.
You can technically make it from Tukwila Amtrak to the airport with local buses (line F west to Tukwila International Boulevard Station - not the same as Tukwila (Amtrak) Station - then line A south to 176th St.), but it will be far less hassle and almost certainly faster to just take the one-seat ride on the light rail from International District Station, one block east of Seattle/King Street, to the airport. 31 minutes travel time and $3.00 fare.