For what it's worth, there are some great things to see and do in Astoria/Long Island City, which is much closer to LGA than Manhattan (i.e., you would have more time to explore). Disclaimer: I live in Astoria!

I would take to Taverna Kyclades for a Greek lunch (amazing swordfish!), then either to the Museum of the Moving Image or the Isamu Noguchi Museum. You'd even have time for an early dinner -- maybe Brazilian? The incomparable Egyptian hole-in-the-wall Kebab Cafe? Or maybe an upscale Italian?

No need to head into Manhattan! And wherever you are in Astoria, it's only a 10-minute or so car ride to LGA.