13 Coins is a classic restaurant and the source of our long-standing family story about Fettuccini Apollo. (long story, not that funny so never mind right now~~~) Fairly high end restaurant. If you are wanting somethng a little more "local", there are a lot of neighborhood pubs etc.
Just so you know...the airport is about 30 miles south of Seattle proper and nothing of interest out there except the usual Holiday Inn.
13 Coins is kind of on the north end of downtown Seattle. Probably enough time on a 6 hour layover to head into the city to 1 local, eat/drink, and then back to SeaTac. 13 Coins is GOOD! we had dinner there a couple years ago. Just dinner, didn't go into the bar. Maybe check out some YELP or other Seattle blogs for other Seattle p[laces of interest.
Kell's Bar is in downtown right near the Pike Place public market. It's purported to be haunted, and it's a cool local place that's been there for a long time. Can't vouch for if it is acutally haunted or not, but we had drinks there a couple years ago....and it's a bar.
And it's right near the waterfront where there a lots of other establishments of all sorts.
Likely you'll get more suggestions than able to explore, have fun wherever you go!