Yah, cabo, there's no easy and cheap
place to duck up the Hudson as you round the Battery
. Overnight at Liberty Marina or one of the other jersey marinas
that are just around the corner, is about the most reasonable bet IF you want to stay in that area.
Heading up the Hudson...Remember it is also tidal, it can take longer to travel the Hudson that you'd expect. By all means, daytrip up and eyeball things, but anchoring
free inside Sandy Hook would be the better choice, if you can time it that way.
There are also a number of easily overlooked spots, like Little Bay in the shadow of the Throg's Neck Bridge, where you can anchor
for free if nightfall catches you where you hadn't planned to be.