Is this your first time in Rio? If so, I think heading into town is a no brainer. If you just want to relax on the beach, I would suggest perhaps getting a day use rate at a hotel in Copacabana, and heading to the beach (an Ibis would do fine in that case, and will be nicer and cheaper than the Linx hotel at the airport).
If it's sightseeing that you're after, and you'd like to maximize your time on the ground, I would suggest something like "rent a local friend", which is a sort of Airbnb for informal local tour guides (might cost around U$100 for 6h): you could arrange to get picked up and dropped off at GIG (assuming your "friend" owns a car, and visit Sugarloaf, have a nice lunch (Rubaiyat or Fogo de Chao if you'd like some steak), perhaps the Olympic port area, or have a caipirinha by the beach Riba kiosk in Leblon or Ipanema would be a must here.
Bear in mind that if it's a weekday you should budget at least 70 minutes to get to GIG from Zona Sul during rush hour, potentially more on a Friday (only 30-40 mins on weekends).
The new AAdmirals Club at GIG is nice enough that you might want to spend some hours there, although not sure when check in would open (assuming you're flying AA).