I'm a huge Japanophile: if there's one other place I'd like to live, it's Tokyo. I must have been there seven or eight times, most recently just after the Tohoku earthquake and tsunami. Anyway, EN is a gem on Hudson Street, serving real Japanese cuisine. It turns out that EN is a chain in Japan; there are a lot of branches making lovely bosky food in cosy neighbourhood locations. But their New York incarnation is grand in scale and ambition, with solid, warm interiors (not unlike if the Whitney were a Japanese restaurant, oddly) - a remarkable hybrid of this city, and the other one that I'd love to live in.
The New York Earth Room is an interior sculpture by the artist Walter de Maria that has been installed in a loft at 141 Wooster Street since 1977. Go see!
The oldest still-operating restaurant in New York City, still in the family of the original founders. At Christmastime, it's decorated to the nines. Go with a big group: portions are large and the wine flows liberally. Look for the little buttons on the walls: in the old days, the mafia would hang out in the back room, and if diners saw the cops come in, they'd press the buttons so the mafia guys could run out the back.