I don't think the airport shuttles come out to Madison, but you can take a NJ Transit train. You'll have to leave from the Madison train station and transfer in Secaucus. The train from Secaucus will take you to the airport. Go to njtransit.com and enter Madison as your "Travel From" station and EWR as your "Travel To" then select "Rail" under Using. You can also enter desired times. It will tell you when the Madison train leaves and arrives in Secaucus and will do the same for the train from Secaucus to the airport.
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