Hi, I'm in northern NJ and I dayhike the catskills all the time... You're biggest problem (depending on which boro you're in of course) is getting out of the city. Once you get over to Orange/Rockland county, you would be anywhere from 1 hour 45 minutes to say 2 hrs 15 minutes from most places in the catskills. I usually use Woodland Valley in Phoenicia as a standard gauge. I am about 1 hour 45 minutes from Woodland Valley from northern NJ so obviously, you have plenty of opportunities for a dayhike in the catskills. Just add that time to how long it takes to get out of NYC proper.
Isn't the White mountains from Boston a good 2.5-3 hours? If so, then you should easily be able to do a bunch of catskill hikes similarly. Don't forget the Gunks, Harriman, East Hudson, West Hudson, CT, and the parks in northern NJ for dayhikes. Not much in terms of good backpacking sites other than the AT or so, but plenty of dayhikes in and around the lower Hudson and northern NJ.. The Gunks would be about 1/2 hour closer than the catskills... of course, depending on where in the gunks..
Jay