To make your rum last as long as possible, keep it in a cool, dark, dry place, whether it's open or unopened. This could be a pantry or a basement, if you have one.
Serious rum connoisseurs might invest in a humidor to exercise complete control over moisture exposure, but well-placed storage can do the trick just as well.
If you've opened the bottle, make sure you seal it tight to limit air from getting in. If it has a cork, store it upright, as the cork can break down if the rum keeps touching it.