I'm a former realtor & NYC area resident my entire life. Unfortunately, I need to tell you the truth.
1. Your budget is too low. If you want a spacious 2 family home in Queens or Brooklyn, you need at least $800,000. You also need cash for a down payment + cash to pay closing costs.
2. Rooms inside New York City 2 family houses are small. A spacious 2 family house should have at least 2,500 square feet of living space. 2,000 sq. ft. means the house is an average size 2 family. Anything under 2,000 sq. ft is tiny. I sold a 2 family house with 3 bedrooms each and 2 kitchens. The house is in Yonkers (Westchester County). The 2 living rooms were small and all the bedrooms were small. The 2 kitchens were small - average size.
3. A 2 family house in the suburbs is cheaper and you could buy one in Yonkers or Mount Vernon for $600,000. Another one is New Rochelle but you will need at least $650,000. You should also try Nassau County and other neighborhoods in Queens. Have you looked at Staten Island? They have 2 family houses that are cheaper than Queens & Brooklyn.
4. You will have difficulty finding a realtor because they need to get paid a commission and once they hear the amount you are pre approved for and buying a 2 family house, they will turn you down because of everything I pointed out to you.