They are plenty of public urban gardens in NYC. Check them out for info.
Oh, throw egg shells into the compost. It'll give it plenty of calcium.
Not only public urban gardens but also rooftop gardens.
Egg shells are great but do take a while to break down and be useful. Dolomitic limestone will provide calcium, which helps prevent blossom end rot and will also "sweeten" the soil. Like anything else you have to see which is more readily available to the plants. It may be in the soil but locked up.
Another good thing to plant is beans. It fixes the nitrogen so it's more readily available.
Many people have the wrong idea about compost though. It's nutrient level is low compared to other types of fertilizer. It does however have more benefits than just the nutrients. I swear by it. It can be used to break up a clayey soil and to bind together a sandy one. (I still wonder about that sometimes though. I've dumped literally hundreds of pounds into my sand and can't tell much of a difference. I use the neighbors front end loader to dump it.)
A nice pile of leaves and grass clippings composting over the winter will provide a great side dressing and mulch. Dried leaves will provide a good carbon source and the clippings a good nitrogen source.
Raised beds are nice too. Just don't use any PT for any part of them.
Most sawdust is good for composting. But stay away from walnut. It has a natural herbicide. Use only sawdust from solid boards. No plywood, pressboard or other engineered lumbers.