Amtrak does not stop in Watsonville, and there is little or no possibility of that happening in the foreseeable future. If Caltrain or Cap Corridor ever extends to Salinas, something might be worked out with that as it's more of a local service.
The Santa Cruz branch at this time is adequate for very low-speed use, but it's not feasible to use it for actual transportation-oriented passenger service; it's OK for local freight and low-speed tourist trains. Consider it advanced rail-banked: operating minimally. If money can be found it can be upgraded. However, for now, the bus services between Santa Cruz and Watsonville cover the transit demand. See: [
www.scmtd.com] and click on the Cabrillo-South County region.
Current bus schedule timings range from about 40 minutes (freeway commuter express) to 1:15 (locals via Soquel Dr. & Freedom Blvd.). Subject to traffic of course so if time is critical add 15 min. or more. In the present state of the Branch, I suspect a passenger train (even just a DMU car) on the Branch would take closer to 2 hours to make the trip.