I'm going to NY in June 2014 while AirTrain will be suspended due to repair work. Is there still any not-depending-on-traffic way to go to Penn Station, NewYork?
I don't think there are any other rail options from the airport itself, so you'll necessarily be on the road (and subject to traffic) at least part of the journey.
The nearest train station in that direction is Newark Penn, and the link you posted includes options for getting there by bus. You could also take a cab.