Its an easy day from Port Washington to Sandy Hook. There is decent priced fuel at Atlantic Highlands and you can take a slip, a mooring or anchor behind the breakwall.
We do an overnight now all the time because its just easier and cheaper. Its about 24 hours to Cape May. Often, when we arrive off Cape May in the morning we can catch the current and run all the way up the Delaware Bay to the C and D canal. Some boats make it all the way to the Sassafras River on a fair tide.
But if you don't want to overnight, there are choices. The first inlet is Manasquan, which is about 30 miles from Atlantic Highlands. The inlet is good but there is no room to anchor so docking is mandatory.
Next is Barnegat inlet. It is dangerous with N or NE wind/waves but has anchoring room. Local knowledge is required to run the inlet.
The next day you can make Atlantic City with a very good inlet. There is anchoring room off the CG station but the current really rips there, it always seems windy all night and there is a lot of boat traffic. Or you can stay at a marina.
Then its off to Cape May. Excellent inlet and decent anchorage or dockspace is available. I'm not sure if you can get under the bridges on the Cape May canal. If you can, that saves a long trip back out in the ocean and around Cape May into Delaware Bay. There is a shortcut VERY near shore inside Prissy Wicks Shoal but you need a bit of local knowledge to do it. Otherwise its a good 5 miles out and around.
A good alternative to Cape May is Cape Henlopen on the south side of the bay. Great anchorage but no dockage inside the harbor.
With a bit of patience, you can wait for good weather at Sandy Hook and have a good trip.
And the run down the East River is excellent!!