Your other cheap options are to take trimet to either Amtrak

Union Station or Greyhound station. And then either take greyhound bus or Amtrak train. Do check the train and greyhound schedule so they'll connect to your flight.

It's not the simplest way. But there won't any cheap simple way to Eugene.

My guess is airfare to Eugene (though not cheap) will still be cheaper than getting a shuttle, if you can find any.