Can I hang a bathroom cabinet on a single stud and drywall anchors?
While having all four mount points connect to structure (aka: the stud) is ideal, I think in your case, having two mount points in wood and two in a drywall anchor, you're going to be ok.

Consider this question:
What is the weight capacity of a drywall screw?

One drywall screw CAN (not should) hold a lot of weight for its size.  Also a properly installed drywall anchor can typically stand 70~ pounds of pull force (as in pulling out, not down).

With two WOOD (not drywall - wood) screws set into the available stud and 2 (or better, 4) drywall anchors, I think you're not likely to exceed the capacity.  If your cabinet total weight when full is &lt; 100 lbs you're probably not going to have a problem.