score:3
The Q70 doesn't go to Manhattan. You take the Q70 from LGA to Roosevelt Ave.
Then take either the E or F to Midtown (Carnegie Hall is on W57th Street) or near Times Square or further Downtown.
To get back, take the E to Sutphin Boulevard, then take the AirTrain to JFK. The AirTrain is $5 more but much faster then the bus.
You can substitute the Long Island Rail Road (faster, somewhat nicer but more expensive) for the subway by taking the Q70 to Woodside. Coming back from Penn Station, any train that stops in Jamaica where the AirTrain is.
Upvote:4
LGA to Manhattan: M60 bus from LGA to Manhattan.
Manhattan to JFK: Take the E line from Manhattan to the Jamaica-179th Station and then the Q2 bus to JFK.
Each one will take about an hour. It looks like you're already deducting the 2h early you're supposed to get to the airport from the window (9am to 5:30pm is 8h30m) so I guess doing it this way would give you about 4h in Manhattan.