The 3D cursor, when moved with ShiftRMB-drag, can be Ctrl snapped, so the settings on the left below will at least align its Z to face normal:
But, as an aside, IMO, the 3D cursor tool is still pretty poorly integrated, so, for instance, the ShiftS snap to selected doesn't respect the tool setting: 'Align to Geometry', among other irritations.
So personally, I always have 'Create Orientation', (the little plus in the right hand panel, above), set in my Q quick favourites menu, with 'Overwrite Previous' and 'Use After Creation' set in its F9 panel, so as not to pile up created orientations.
That has the advantage that you can create a Z direction from a face or any 3 vertices, or a Y direction from an edge or any 2 vertices. (Excess defining selections will use a mean direction.) The keyboard , comma shortcut can quickly restore previous transform orientations.