Mondays are OK.
Let's have a look at some of the museums open Mondays -
• American Museum of Natural History
• Jewish Museum
• Museum of Modern Art (MoMA)
• National Museum of the American Indian
• Neue Galerie
• Guggenheim Museum
• South Street Seaport Museum
Any of these museums could be paired with a nearby restaurant or bar, making a complete full afternoon or day in New York. Monday is especially good for a museum visit, because the crowds tend to be thinner, and restaurants, too, tend to be less busy than on a weekend.
A fun museum and bistro walk on the Upper West Side would be a combination of the American Museum of Natural History and the nearby Cafe Lalo on W. 83rd St. I also would suggest a pairing of the Neue Galerie with a nearby cafe, but the two cafes inside the musuem are so good, why go anywhere else?
Image above: The Guggenheim on left and Beaux-Arts townhouse on right. View from E. 88th St. by Walking Off the Big Apple.