There is no sugar in straight rum, although there may be added sugar in flavored rums or in rum-based liqueurs. The liver does not metabolize rum or other types of alcohol into sugar.
Although rum is often distilled from sugar and molasses, the end result contains zero sugar content. After going through the process of fermentation and distillation, rum contains ethyl alcohol, which has no carbohydrates that can be metabolized into sugar. The liver breaks down alcohol into acetate, and eventually into water and carbon dioxide. Drinking distilled alcohol has been demonstrated to actually lower blood sugar levels.
Manufacturers may sometimes add sugar to rum, either in flavorings or as caramel for color.