Here are some thoughts and questions:
1. Do you want the current owner to stay? In general, it is not the best practice to let the previous owner stay at the property for any reason.
2. I know there area and it IS a great deal.
3. NYC evictions are the worst! Try to buy the property predicated on it being delivered vacant. That will allow you to rehab without any problems with tenants in place. Unfortunately, that tends to hurt the cashflow so you'll need to have enough cash on hand to hold the property.
4. Roof and boiler for a 4 family will run you about 25K. However, if you get the property vacant and you have $, then what I recommend is what all the new flippers/renovators are doing. Remove the boiler and install split units that do heat and cool. Also, install electric hot water heaters that are hooked up to the tenants utilities. This will reduce your overhead AND it will allow you to attract better tenants. I know it's pricey, but to be honest if you're getting that price on the property it's worth it. You can rent it out to some hipsters that are never home. Also skip the property management there's nothing they will do that you can't for the amount of money they will charge.
5. Are you thinking of living in the property. Here is why I'm asking. You cannot legally rent out the basement unit. If you tell the bank that, they will not lend. You can however renovate the building and AFTER inspections are done, then fix up the basement to live in it yourself. This is less than 'legal' but is common.
6. 6.3% at Chase is BS. This being your first deal makes it a gamble but if you're really able to get the building at that price you can borrow hard money for the purchase and renovation. Like I said, it's a gamble but at the $350K it's worth it. You're probably looking at 8% interest. If you borrow a total of 450K you will have a completely turnkey property that will rent out easily. When you refinance at a reasonable rate of 3.25-3.75% the financial variance between borrowing $350K or $450K will not be significant. I'm going to tag @John Hickey in this as I have found him to be super knowledgeable and realistic when it comes to all this.
7. Lastly, I'd suggest you be a bit more conservative on your rental estimates. If it's in East Flatbush then that's better than Brownsville. If you're by the 3 train, that's better than not. However, I don't think a 1 bedroom will go for $1600 around here.
8. Ok, now lastly: check for HPD and DOB violations. These can sneak up and screw you royally.
Most importantly, good luck! Let me know how this pans out, $350K for a 4 family (legal) is a great price. If you pass on the deal let me know!