I wouldn't want to anchor in Flushing Bay unless I had to. With planes taking off and landing every minute I'd never be able to get to sleep.
Maybe he anchored off of City Island in the western Sound. The #6 subway is reachable by bus from there. The Liberty Island anchorage is also a possibility but I'd check with the local authorities. I'm fairly certain that you can also anchor above the 79th St. Boat Basin below 125th St or so.
Not much is 'for free' these days when it comes to boating. The only time I have ever stayed at a marina in this area without paying was by arriving late and leaving quite early but that's a bit of a shabby practice (in at 8, out at 6).
The current of the Hudson up at the 79th St. Boat Basin is considerable at around 2 - 3 knots. It is a pretty convenient spot for visiting Manhattan and seeing some sights though. NYC Parks Dept. also runs a small mooring field below Pier 40 that is a bit less exposed to the river: Hudson River Park - Pier 40 Moorings
and is about 5 blocks from where I am. There are only 16 or 18 moorings and they seem to try to get seasonal residents for them instead of transients, er, visitors. Right now this small mooring field is less then halfway filled. If you find one of their buoys to be vacant and the lines are a bit dirty it might be a "free" mooring. NYC Parks Dept. moorings are about 35-40$/night at either Pier 40 or 79th Street.
Another option might be to get a 'transient' mooring at a club in one of the western harbors of Long Island, like Manhasset, Hempstead, Oyster Bay. Oyster Bay has a LI rail road station right by the waterfront. The train ride from there to 34th St/Penn Station in NYC is over an hour long but it is a much safer anchorage in Oyster Bay then the Hudson River with it's strong reversing current. In Oyster Bay the tide gently goes in and out with a 6'+ tidal range but the current is 1/2 knot or less. The same should also be true of most harbors on the Connecticut shore of the LI Sound regarding the current being gentle.
Let us know how you fare.