This is geometry question about subtending an angle.
First of all, let's figure out just how big our sunshade is. Looks like NYC is 784 square kilometers or so. Let's assume it's circular, which lets us use a bit of math to figure out that it's 16km in radius, or 32km in diameter.
If we're directly right under the middle of the umbrella, we can look straight up and see its center -- let's call this looking up at 0 degrees (if we look at the horizon, that'll be 90 degrees.) If we look over to the edge of the big black dot in the sky, we're looking at some angle between 0 degrees and 90 degrees. With a fixed radius, this angle is purely a function of altitude.
Let's start off with a nice equilateral triangle, so we look down 30 degrees from up (or up 60 degrees from horizontal) and see the edge. We can calculate an altitude of 27.65 kilometers.
Just how shady is that? Well, the earth rotates such that the sun takes about 12 hours to move across the 180 degrees of the sky, which means that it seems to move 15 degrees every hour. (This is also why a timezone is 15 degrees of longitude wide.)
Assuming it cuts across the middle of our sunshade, it takes 4 hours to traverse the 60 degrees from one side to the other. The other 8 hours of the day we still get the sun.
To solve for altitude given a desired X hours of shade, we can arrange our formulas to:
Altitude = cosine(hours-of-shade * 7.5 degrees) * 32 km
8 hours of shade is 16 km up.
A crippling 11.5 hours of shade is just 2 km up.
For reference, low Earth orbit (LEO) is usually considered to be between 160km and 2000km up. If our floating city is 160km up then it's not going to be very shady for very long -- I'll leave the calculation as an exercise for the reader.
For a sense of scale, this is like looking at a standard schoolbus that's a football-field away.
Hope that helps!