This code will add a 15x15 icon in the upper-right corner of the event. Unless you have ".event_action_menu" defined in your CSS it won't be visible (it's defined in Demo/Themes/areas.css).
1. Try using hard-coded red color to make sure it's visible.
2. Also, add Visible() call to make sure it's permanently visible (not just on hover).
3. You need to use ClientObjectName property of an actual DayPilotMenu object when specifying the context menu action.
e.Areas.Add(new Area().Right(20).Top(3).Width(15).Height(15).BackgroundColor("red").Visible().ContextMenu(DayPilotMenu1.ClientObjectName));
Then it will be possible to activate the context menu by tapping this red icon on your phone.