I moved here about nine months ago with no job and no apartment in place. I had more than $3k saved up, but I'd say between the time I arrived and receiving my first paycheck, I spent only about that much (and it would have been significantly less if I hadn't had to pay three months of rent [first, last, security] for my room-- most places seem to only require first and a security deposit). I also live off the L, kind of straddling the divide between East Williamsburg and Bushwick.
How I did it: I spent the preceding months doing research on neighborhoods and narrowing down the ones that both fit my estimated budget and sounded like somewhere I wanted to be. Bushwick was pretty high on that list, though friends of mine who lived here suggested places like Astoria, Ridgewood, Sunnyside. I would look at room share listings on Craigslist to get an idea of what the average rents were to get a room in various neighborhoods.
As far as jobs went, I did not bother trying to apply until I got here. I knew that places would toss my resume immediately, and even if by a miracle I got an interview, I could not afford to travel all the way to New York just for that. It was better to save my money so I could stay here longer. I did make sure to have my resume updated and a cover letter prepared before I came.
I quit my job, downsized my possessions, sold my bed, didn't have a car so didn't have to worry about that, packed two suitcases with my essentials, and whatever didn't fit went into boxes to stay at a relative's until I could afford to have it shipped to me in New York.
Living arrangements: I did already know some people who lived here, but no one I was close enough with to feel comfortable asking to crash with even for a few weeks. I booked two weeks on someone's couch off AirBNB for $15 a night. It was like $240 or something for the whole two weeks. (After I came, one of my friends did offer to let me stay with them if I hadn't found something permanent by then, but thankfully it did not come to that.) The apartment was actually in a fantastic location on the UWS, but the first two nights I slept(/squeezed) in a tiny makeshift loft that was really just a hole in the ceiling, it was pretty awful, then I transferred to the couch, but the apartment had no kitchen, the guy who lived there was… kind of off, he basically operated it like a hostel so there were like six girls around. I think AirBNB has since pulled him as a host since too many "guests" complained. Whatever. I'm pretty good at rolling with the punches and none of it bothered me too badly. I wasn't there for a vacation, just somewhere to crash while I tried to figure something permanent out.
The Craigslist research I did before I moved helped me figure out what rooms I'd be interested in, and ones that I knew would not require a credit check/job verification. I even started inquiring the day before I left so I could have viewings lined up. I flew in one evening and literally the next morning was looking at an apartment that was advertised as being in Bushwick but was more in Bed-Stuy. That place was really shitty and I didn't have a good feeling about the roommate situation. Sent more inquiries, lined up two more rooms to look at the next day, saw the one in East Williamsburg and immediately knew it was what I wanted. It was on the higher end of my rent budget, but a steal for the neighborhood and fit all of my criteria. Even better, it was partially furnished with a bed and some shelves, which was a big plus because buying a whole bed set would have been difficult and expensive (a fair amount of room share listings do come furnished). Wrote a check on the spot and had several panic attacks after just because it was really sinking in.
Since I couldn't move in until the start of the month, that left me about a week and a half to sort everything else out. I waited until I'd been there a full week before looking for a job (I wanted to make sure I had somewhere to live before doing that); I don't have a college degree, but I have a lot of solid work experience for my field, and I was lucky to have three interviews within a few days of applying. At the second interview I was offered the job right there. The quickness in which I found employment is obviously not what a lot of people experience when they move to the city, and I'd been prepared to have to slog it out a lot longer.
How to live cheap: One thing I learned is that for most things, New York actually DOES provide a cheap(er) alternative; the issue is really how willing you are to go out of your way for the sake of saving a few bucks. I looked up a lot of places to find the cheapest ways I could buy necessities like pillows, hangers, etc and feed myself. It meant trips to Lots-For-Less and Jack's (discount stores with prices comparable to midwest Wal-Marts) and eating dollar pizza for meals many times. I got a free Ikea comforter off Craigslist. I bought cheap groceries instead of eating out all the time. I moved all of my things to Brooklyn via the subway, which meant probably five separate trips and was totally miserable, but saved me a lot versus paying for a cab or a moving truck.
Anecdotes are not empirical evidence, and I get why people caution so much against people moving here without a big safety net. But it isn't impossible to do. It depends on how willing you are to sacrifice to make it happen. I'd suggest that while you're here on your trip, get an unlimited week Metrocard and take the L down and get off at different stops and just walk around a bit so you can get a feel for each neighborhood.