Can I get there from New York City without a car?+
Yes, more easily than anywhere else on this list. Trains run up the Hudson line to Beacon, Cold Spring, Rhinecliff and Hudson, and buses serve the main Catskills towns. From a station you will still need a taxi or a pickup for the last stretch, since almost nothing is walkable from a platform. Ask your host about a shuttle before assuming.
Is two hours from the city far enough to feel away?+
Usually yes, and the deciding factor is phone signal rather than distance. Deep in the Catskills the valleys cut coverage to nothing, which does more for the nervous system than another hundred miles would. The Hudson Valley proper stays connected and social, pretty but not remote. If you want genuine remoteness, go north to the Adirondacks instead.
How bad are ticks here really?+
This is one of the highest Lyme disease regions in the country and it deserves plain attention rather than worry. Ticks are active any time the ground is above freezing, and they sit on tall grass at leg height rather than dropping from trees. Stay on trail centres, tuck trousers into socks, use permethrin treated clothing if you can, and check yourself every evening. Found and removed within a day, the risk is low.