In order for you to find the answer to your question you need to have a continuity tester, or some type of meter that can measure ohms. You could also use a plug continuity tester
First make sure all circuits are dead in the box you are testing, then see if you can get a "ring" between the neutral and the metal box (make sure they are not touching in the box). If you do get a ring then chances are your system is grounded through the metal conduit or cable back to the panel. If it is grounded to the panel then you can go ahead and replace the receptacle. Is this a good idea? Probably not, but if your dwelling falls under an old grandfather clause that uses the metal conduit as a ground there is no reason you can't.
The more modern and acceptable approach or if you can't get a ring between the neutral and the ground, is to replace that receptacle with a GFCI receptacle. If all of the outlets on this circuit are the same configuration as the one shown. Then the best method would be to find the beginning of the circuit and install a GFCI there and label the preceding receptacles ungrounded with the little labels provided with the GFCI receptacle.
Here is a picture to help explain, just remember the is no ground.