I'm not sure how Staten Island became part of New York City (and I'm not going to bother researching it). It sits below Hoboken and Jersey City, which feel like they could be more a part of New York City than Staten Island. I've buzzed across 440 and I-278 several times taking my wife back and forth to Brooklyn. Other times I've crawled across I-278 to the point I said "this traffic is so bad, lets take local roads all the way across Staten Island to the Outerbridge Crossing" (which I never knew the name until now). That only confirmed my "this feels like New Jersey" mood of Staten Island.
On a drive taking my brother back to NYC I convinced him to make a stop in Staten Island to drive past the apartment complex where Method Man grew up (because why not). You pull up and hey what do you know it's an apartment complex. I have these photos someplace (I have an Instagram post of the place) and need to sort through a few photos to find the images, although my phone that had these images died. The place didn't seem to rough and tumble but who knows what it was like thirty years ago.
We then drove down to Flagship Brewing. Then on the drive back to I-278 we stopped for a few photos of the NYC skyline - not as much of a closeup as Brooklyn provides but still a great shot of Lower Manhattan.