I HIGHLY recommend against running any appliance, no matter what it is, that is intended to be grounded, on an ungrounded circuit.
Furthermore, even if that screw-in DID have a ground, you would still need to check the max load that it can safely handle (in amps) against the rated draw of your light. If your light draws more than 85% of the rated max load on that circuit, you are SOL.
Your setup sounds a little frightening, frankly. Be very VERY careful not to overload that circuit, and since apartment storage areas are not necessarily your 'home', your rights may be very different, and if it runs off the main breaker box for teh complex, instead of on your personal bill, the landlord might start sniffing around looking for something fishy.