For the Avenue that wears its age with grace and not a lot of show, I give the prize to Fifth Avenue in Brooklyn, from Boerum Hill on down through Park Slope to Windsor Terrace.
There are very old bars on this strip that do not get a lot of play in the papers—places like O'Connor's, Smith's, Jackie's Fifth Amendment and Smolen Bar & Grill. Likewise, you'll find a lot of retail stores and diners which are quietly marking their decades, such as Eagle Provisions.
Here are a couple of long-serving slice joints which sit a few blocks from each other: Luigi's (since 21st), which was founded in 1973; and Lenny's (near 17th), which dates back from the 1950s. You can see what Lenny looked like it you walk on to the back. There's an oil portrait of the man in a suit, complete with boutonniere.