From the Cold Spring platform, you can do a solid car-free day, but you have to separate the two things people mean by "hike."
Little Stony Point State Park is real and walkable, roughly 20–25 minutes on foot from the station. The easiest route is not just hugging 9D the whole time. From the platform, cross into the village, walk north on Main Street, then pick up the riverfront route near the bandstand/waterfront. You'll reach the park entrance at about the bend where the road crosses the train tracks. Inside the park there's a short loop up above the Hudson; it's more like a nature walk than a real trail day, but the river views are good.
Actual trailheads for Bull Hill, Washburn, and Breakneck are a different story. Some are approachable from 9D, but that stretch is a 40–45 mph road with narrow shoulders. I've done it, but I wouldn't call it pleasant or particularly safe on a summer Saturday. There is a seasonal shuttle/trolley thing some years, but service isn't guaranteed. If you want a longer hike, I'd start at Little Stony Point, then head back into the village for lunch, and save the real trailheads for when you've scoped out the 9D walking conditions in person.
Weekend trains back can be hourly-ish, so pick your return time and don't miss it. Main Street food is steps from the station. You don't need Uber for the basic version.