Near the 4th Ave address you listed, there is the Rock Bottom, a pub style restaurant with good food. They're at 1333 Fifth Avenue. Sazerac Restaurant is good with a southern style menu, right at the address you listed.
Further up on 2nd Avenue, there are lots of options as you are in Belltown. One block to the west on 1st, there is the Flying Fish, the Queen City Grill, Jai Thai, Ohana (a really good Hawaiian restaurant) and a BBQ restaurant right next door. A couple of blocks down on 4th is one of my favorites, the Dalia Lounge, as well as my favorite steak house, Buenos Aires Grill, an Argentinian steakhouse on Virginia between 2nd & 3rd.
Lots of choices, depends on what cuisine you're in the mood for and how much you want to dent the pocket book.
Peter