Lizzie, if you want to do bottom paint, you need to check with the yard. A number of yards up in New Rochelle (Echo Bay) stopped allowing bottom painters years ago because the EPA or NYS can shut them down if the paint or removed paint dust gets in the ground and they are declared toxic waste sites. They're scared, for good reason.
Most yards will still allow bottom paint, but DIY hull paint is less likely because they are worried you'll get overspray on the next boats and folks will sue for damages. Again, you need to check with the yard. Gateway Marina (aka Dead Horse Bay) at the south end of Flatbush Avenue, across from Floyd Bennett Field in Brooklyn, could work for you. A short bus transfer or stiff walk from the F train, informal relatively cheap, you'll need to watch the charts and shoals to get in/out though, but there isn't much in NYC that has both yard space and transit access.
Oyster Bay Marina is almost at the Oyster Bay LIRR yards, an easy commute (40 minutes?) by LIRR and good deep water but I have no idea about allowing work in the yard. Could be faster to reach than City Island, which requires a long subway ride and then another bus transfer, for most of NYC.
If the transit time to City Island doesn't bother you, ask at the yacht clubs there. Many are old and informal, yard space is tight but you're joining a real club not just storing a boat. Could be worth the day to go up and walk around.
And then I think there's one called Liberty State Marina (not sure of the name) just north of Libetry State Park in NJ, which is supposed to be reasonable and easy to reach via PATH train if you're starting out from parts of Manhattan. A nuisance double-commute for the rest of NYC though.
I think that's all you'll find convenient to NYC that meets your requirements for yard space and transit though. Don't know about further out on the North Shore.