You just take the train to union station and then Uber or the hotel will pick you up from there. It's only 7 blocks, we walked it.
The hotel does have an SUV that they will drive some people around in (not certain of criteria) though doubtful all the way to the airport (not sure). Best bet would be to use that SUV to go to union station then take the train straight out to the airport... If they wont take you to union station even then the 16th st mall bus line runs to union station from a half block walk away from the hotel entrance (for free).
I'm not aware of shuttles. I took a Lyft.
Thanks! Your answer is awaiting moderation.