firstwinter, one small correction: going north from Grand Central, the next stop after Breakneck is usually Beacon, not Cold Spring. Beacon is a bigger station and actually not a terrible bailout, but it is a detour, so don't assume you can just hop across the platform and double back immediately.
For the original question: Breakneck Ridge station is not on every train, even on weekends. Check the Hudson Line schedule for trains that list it, because some pass right through. In my experience, the conductor announces the stop and only opens the correct car—usually the last car heading north, but listen for the call. The station itself is very minimal: short platform, no bathroom, no staff, no waiting room. You walk from the platform along the road to the trailhead.
The main white-blazed trail goes up fast. For a generally fit beginner, the issue isn't distance—it's that you'll be using your hands on steep exposed rock. I'd call it a scramble, not a walk. If it's wet or your shoes have no grip, save it for another day. A reasonable plan is to go up the white trail to the first big viewpoint, then take the yellow/red or blue trail down and walk back along 9D to the station. That avoids descending the steepest part, which is where most trouble happens.
Solo on a decent-weather weekend is fine; there are usually enough people around. Tell someone your train times, and leave a buffer for the return, because service back can be sparse and the platform has no shelter.