I was EWR-based for a long time, although I commuted. I never heard of a direct shuttle between EWR and LGA. There is (or was) one between EWR/JFK.
There's the Newark Liberty Airport Express (
http://www.coachusa.com/olympia/) that runs directly between EWR and downtown Manhattan every 15 or 20 minutes or so. I think it makes several stops, including Grand Central Station. From there, you should be able to catch the subway or a bus to LGA. The tickets are around $10/each way if you buy 10 at a time with an airline ID. I know of several CAL and XJT pilots/FAs that live in NYC and use this bus service.
Good luck.