You're NUTS!! As a former realtor and NYC area resident, a normal size 2 bedroom, laundry room and heat / hot water = $4 - 5,000./month in the nice / safe areas and about $3,000./month in low income areas. A small 2 bedroom = $3,000. in the better neighborhoods and about $2,000. in the low income neighborhoods.
Your budget of $1800. was last seen in 2008 - 2010. It's now 2017. You also need to learn that utilities are never included. Landlords cannot control what a tenant spends on utilities so therefore the tenant must pay for it separately.
You should try looking in Nassau County. Rents are much cheaper than Brooklyn. There is excellent public transportation - Long Island Railroad and public busses - they do go into Brooklyn and Queens too.