I am in New York and I need to go to Arizona, but my passport has expired. Can I still buy the plane ticket or should I just use my green card? Note that I do not have a driver's license that may be used as ID.
You can use your permanent resident card ("green card") as evidence of your identity for the purpose of domestic air travel within the United States.
Note that you must show "valid identification" (emphasis mine). I would take this to assume that an expired passport is not a valid one. However I can see the word is ambiguous.
Yes, you can do this. I travel regularly from San Francisco to New York and Boston without taking my (EU) passport. I need only show driver's license, and even that isn't, strictly speaking, necessary.
For travel within the U.S., you have to show identification. The list of acceptable forms of identification is available at https://www.tsa.gov/travel/security-screening/identification. A passport is one valid form of identification, but certainly not the only one. Personally, I have never shown my passport to fly: I always use my drivers license. I flew for many years before I got my first passport.