Basically, its shelf life is indefinite if stored properly. As you know by now, after opening the bottle, the rum starts to slowly lose its taste (depending on how it's stored, how much liquid is in the bottle, etc.). Because of that, it's recommended to drink the rum within a couple of years after opening.
Can old rum make you sick?
Can expired alcohol make you sick? Liquor does not expire to the point of causing sickness. It simply loses flavor — generally a year after being opened. Beer that goes bad — or flat — won't make you sick but may upset your stomach.
How long can you keep rum before it goes bad?
Rum will retain both its content and drinkability (for up to 6 months) if the bottle/container is sealed and relatively evaporation free. Contrary to popular belief, alcohol does go bad. So you can't wait too long to finish an opened bottle if you want to savor its real taste.