In my new apartment there are some wall outlets that are two prong outlets. Also my fear is that the three prong outlets are not “physically” connected to ground. I heard somewhere that in this kind of system the neutral wire can also be used as a ground. Is this correct? Would a separate neutral wire need to be run at each outlet for the ground or would the existing wire be able to connect to both the neutral and ground part of the outlet?

If this method is insufficient (or unsafe) are there any other suggestions or tips? After work I am going to take apart one outlet in order to see if there is a ground connection screw attached to the metal casing.

If you have a metal box connected to the fuse box by either metal conduit or flex, it would be much better to rely on it providing an independent path to ground than using the neutral. Best way to check to see if the box has a good ground is to measure the voltage between it and the hot wire. Should be the same as hot to neutral. If the box is grounded, you can install a 3 prong outlet, or use a properly installed 2 prong adapter, putting the little tab under the cover plate screw.

The ground wire connected to the screw is a newer, more secure system. For many years 3 prong outlets were just screwed to metal boxes grounded through metal conduit or flex. Conduit and flex can come loose, or corrode breaking the ground path. Both plastic boxes and conduit are common now. They require a separate ground wire. Lack of a wire, doesn't mean no ground. A metal box and conduit doesn't mean it is grounded. There is enough room for doubt that the outlet checkers are a common item.

Labman,
Thanks for the advise. I will test the outlets tonight for my next move. For testing I'm just going to take my multimeter and measure the voltage from the screw to each slot (one at a time). I should read 0v from one and the 120v from the other if the outlet is grounded correct? If it is grounded like that can I just throw in a regular 3 prong outlet? What would connect to the ground portion of the outlet?

If there is a load on the circuit, You may see a couple of volts between neutral and ground.

3 prong outlets have the metal frame, the part holding the screws to the box and cover plate on connected to the third prong. The frame, including the cover plate screw, of any outlet screwed to a grounded metal box, is grounded. Newer ones have a screw on the underside to hold a ground wire. Code, and devices constantly change trying to keep electrical service both affordable and idiot proof. Using plastic boxes and conduit or NM, you must have a separate wire for a ground. It is also more secure even with metal boxes and conduit.