Let's see how much energy a person needs, so we can see how many people we can feed with our food, assuming we can grow food.
Daily needs for an adult are about
3500 kcal according to this calculator (it uses large calories) for a male (because there's higher needs),
32 years old,
1.80 m tall and
70 kg in weight - this other calculator confirms this, giving
15.62e3 kJ which is the equivalent of
3731 kcal. Let's average those and round to get our daily needs, per individual are
15e3 kJ or
Assuming they live off roof-top crops, they'd be essentially living on a vegan diet and possibly on a raw vegan diet as well. Now, I'm going to assume they eat and grow only soy for simplicity - I'm handwaving the dangers of eating only one kind of food, especially if it's soy - variety is necessary. The outlook is pretty bleak though: going by existing rooftop cultivation examples such as this guy's, it seems you need at least 3 months before you get food out of this - you also need enriched soil, many kinds of plants to make sure you've got food all year round and there's that looming nutritional variety problem. Even those that have done this kind of thing, don't replace all their intake needs from these gardens. But those problems can be handled narratively - lets see if we can force enough food out of soy for the time being.
According to this chart (it's badly made, the lists are images so you can't search >:( )
100 g of soy beans contain
545 kJ boiled. That means we need about
2.75 kg of soy beans, per person, per day. The wikipedia article on soybeans gives us an average yield for soy bean crops of
2.5 tonnes per hectare in 2010. I'm assuming that's the total over a year, but I'm no farmer so I can't be sure. Since that's for a professional, well-tended crop with modern technology, lets cut that yield to 1/3, giving us about
0.8 tonnes per hectare. Converted to
0.08 kg/m^2. For a single person, we need
979 kg of soy beans per year, which requires an area of
12.24e3 m^2, or a square
110 m on each side. Going by this paper, the total flat roof area in NYC is
21,249 acres divided over
144,832 roofs. That's about
342 m^2 per roof, which means one person, over the course of a year, needs almost
36 roofs to feed themselves, just by roof yield, so that doesn't work. It can still be a good supplement, but you'd probably want to find some way to grow food within the buildings themselves, like this:
which is known as vertical farming. If, however, you can find other sources of food, such as scavenged food or make a cat farm, you'd need about
0.62 kg of food per adult, per day (if you're an astronaut at least). That amounts to
220 kg for an adult over the course of a year, excluding water. You should be able to get this much if dietary intake is varied, especially if it contains meats and dairy products (since there is dried meat, jerky etc. left over in supermarkets, we can assume they get some of this, even while scavenging in the city). The less varied the food is, the more you need.
How many people can survive this way would vary largely. It depends on what food you have available, how many buildings are close-by, how much food is scavengable and in what area. It's hard to make an estimate, but I think the biggest factor by far would be scavengable food by area. As far as I can tell, there just isn't enough yield on these crops and too small an area to get enough to feed an entire group of 4-6 people. It would take tons of work for even one person to manage alone, with just their crops - not to mention farming is a huge amount of work, even on a rooftop.
Rope bridges are fairly straightforward, but you'd want someone on the other side, or it can be very tricky to get one to work. I don't know much about how to make them and I can't find much information on it, but as far as I understand, you'd need enough rope to cross the gap one time (with a noose), then add 2 gap lengths if you want to be balancing on a single rope, with 2 side-ropes for handling. You'd probably need 4+ gap-length's of rope to get the simplest of board bridges, possibly double that length.
The distance between buildings can be estimated based on street distance.Going by this NYC street design guide, it seems we'd probably have a minimum of
20 ft and probably a maximum over
72 ft for large streets. If you want to connect buildings close by, with just a side path in between, we're probably talking about as little as
16 ft of distance. Of course, some buildings may be so close you don't need anything besides your legs to cross over.
If we want to create a suspension rope bridge, we'd probably need at least
64 ft of rope and some boards. That's
19.5 m. You would also need some powerful anchors, but that might be easy to find if there's pipes, chimneys etc. - it would depend on what's available.