There's no direct bus service from JFK, you'd have to go to Port Authority Bus Terminal. The Metro North-Port Jervis line train from Penn Station stops at Salisbury Mills/Cornwall or the Hudson line train from Grand Central to Beacon, these are the closest stations to the Newburgh/New Windsor area. Beacon is across the river from Newburgh and New Windsor is just south of Newburgh. NYAS has frequent express bus service from all terminals to Port Authority, Grand Central and Penn Station. These are also accessable by subway.
There's no direct bus