As seen below, Cody got stitched up after the show and AEW says he suffered a costochondral separation. That is sometimes known as a rib separation and can affect your breathing. That type of injury can take 2-3 months to completely recover from.
Rhodes' mother, who was ringside for the match, is also a nurse so she helped to stitch up her son after the match.
Per his promise made before the match at Full Gear, Rhodes can never challenge for the AEW World Championship again. I would assume that his next big pay-per-view match will be against MJF.
During the post-show media scrum, Tony Khan confirmed that there are plans to run a pay-per-view in the first quarter of 2020 so that could be when we see the first MJF vs. Rhodes match.
In the video below, Khan spoke about Rhodes' injury during the match with Jericho.