no way you'll get 20 minutes for grand central for $1000. Park Slope in Brooklyn might be able to fit you for $1k, but it'd be 30+ minutes in the morning. Otherwise Sunset Park might work. Not altogether safe, but if you get near the W, it's 25-35 minutes to grand central, depending on the 7. It's really close to Bay Ridge (lot's of good little bars) and a really mixed neighborhood, with Jewish, Mexican, Asian, and White in pretty equal numbers.
Maybe somewhere near Columbia. (between broadway and amsterdam, 116 plus or minus 5 streets). Neighborhood isn't the safest, but you can find some great aparments for cheap. Just stay really close to the school. And a recent grad would fit in there.
Or check the upper east side, you might get lucky, maybe you'll be able to sublet for that much.
One last place to check would be Roosevelt Island. It's lot's of families, but really close to the city. And you could take a tram to work.
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