A coach from Shenyang to Dandong will take about 3.5 hours; the train takes longer, so there's little point in using it. Unfortunately, there is no direct coach service from SHE to Dandong, so you'll need to get from the airport to the main coach station first. A taxi from SHE to the coach terminal should be about 50RMB.

From the Liaoning Province Express Coach Terminal (辽宁省汽车快速客运站) coaches depart for Dandong every 30-60 minutes between 0700 and 1900. You shouldn't have any trouble buying a ticket at the station. The coach ticket itself is about 70RMB.