Juan Perez what neighborhood are you talking about?
I admire your energy, but you need to get your ducks in a row first.
1. Talk to a mortgage banker/ mortgage broker to get pre-approved for an FHA 203k. If you need a referral for a mortgage guy PM me. They will tell you how much you can borrow in total given your income, debt, and credit score. This should be the absolute first thing you do.
2. Once you know how much you can borrow, then you know your absolute max budget. Remember to factor in closing costs which can be very high in NYC. A good rule of thumb is 3-4% of purchase price. So if you purchase a 2-fam for 400k, your closing costs will be $12-16k. This in general will need to come out of pocket and paid at closing. It's difficult to finance your closing costs here so don't count on it. This means if you buy a 400k property you will need 16k for closing and 3.5% of 400k = $14k for down payment. So right off the bat you need $30k in cash just to close the deal. The bank will likely want you to have reserves also in addition to down payment and closing costs so you might need to have closer to 40k total cash to get started.
3. Once you've completed steps 1 & 2, it's time to go shopping. My thinking is it's going to be hard to find a 2-family for 400k, but if you find a fixer upper, make sure the total cost of the purchase and renovation fit within your max loan amount. Unless you're doing a major renovation (100k+) it's unlikely you are going to create enough appreciation to successfully flip the house in 3-6 months given all the carrying costs and transaction costs.
4. When you locate a property it's time to get quotes from licensed and insured contractors that are willing to work with the FHA payout schedule. Make sure you find a few that are familiar with the FHA program.
5. As folks have mentioned above, FHA requires you to live in the property for 1 year before you move out so make sure you intend to do this. In theory you could refinance out of the FHA loan in 90 days but at that point you'll need 20% equity in the property to switch to a conventional loan which is unlikely to happen.
I think your best bet is to find a fixer upper that doesn't need anything major/structural. Just new kitchens and bathrooms, coat of paint and maybe redo the flooring. Make sure you fix the outside of the house nicely. Move in with your 203k, rent out all of the spare bedrooms or extra apartment for cash flow, and in a year or so, assuming the market is still rising, sell it for a profit. Or you might find you can get positive cash flow renting the bedrooms out separately to college students etc. for a longer term buy and hold.
Good luck!