No easy way. I would uber or lyft over to Metro North then follow above. Going back & forth to New York City is crazy backtracking.
You may be able to use Shortline bus they travel upstate NY as well as NYC. And New Jersey
I don't think directly. You could take the shuttle down to nyc then the metro north up to poughkeepsie, bus over to new paltz trailway station and there may be a bus going to catskill or nearby from there. Whatever way you take public transportation it's going to be inconvenient to go from Stewart anywhere except nyc
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