The Peachtree Center MARTA (train) station is with walking distance of the hotel and that will take you directly to the airport.
No free shuttle. Agree with Ken Mac about hotel room quality.
Was there last month there is no shuttle and I took Lyft which was $13.00 If i had know that the hotel was run down and the rooms were so small especially the bathroom. I would have stayed at a hotel near the airport it probably would have been a better facility and cost less. I paid $331.00 for a single room that was almost the size of a walk in closet.
It is downtown Atlanta, so about 8 miles. I'm certain they don't have a shuttle, but you can take marta to the Peachtree Center station ($2.50) or Uber or Lyft there for about $15. Taxis will probably be $20-25.
I have no idea about the airport bit there is not a free shuttle there is an electronic car kinda like a train you can ride but its not free you have to purchase tickets before you get on but its cheap i think its a dollar
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