Identify the similarities and differences between the Old Kingdom statue of Menkaure and His Wife (c. 2500 BCE) and the Archaic Greek New York Kouros (c. 600 BCE).

By “the Archaic Greek New York Kouros,” I am going to assume that you are referring to the kouros at the Metropolitan Museum of Art in New York. Kouroi (plural of kouros) were free-standing sculptures of young men made during the Archaic period of ancient Greek art. Those depicting women were known as kore, plural korai. They were greatly influenced by earlier and contemporary Egyptian sculpture such as the sculpture of Menkaure and His Wife.
Some similarities between kouroi/korai sculptures and the sculpture of Menkaure and His Wife (also known as Menkaure and a Queen) are that both are nearly life-size, facing frontally, and are taking a step forward. The arms are also held firmly at the sides, although you will notice that the Queen is actually holding Menkaure.
Some of the differences include that kouroi sculptures were generally depicted in the nude, which emphasized their youth and athleticism. This was not the custom in Egyptian art. Menkaure and the Queen are certainly clothed. You will also notice that the Greek kouros youth is depicted with hair—long rows of pulled-back braids—while Menkaure and the Queen are depicted with royal headgear. Overall, Egyptian sculptures, especially those of royals, tended to be a bit stiffer, exuding power and commanding respect. For this reason, Menkaure and the Queen exhibit more rigid postures than the Greek kouros, which seems a bit more relaxed and playful.
Along with the websites below for further reading, I would also recommend Madelynn Dickerson’s The Handy Art History Answer Book.
http://ccivcopy.site.wesleyan.edu/project-6/kouroi/