I know it might not be up to NEC code BUT as I see it: If your wiring does not have a separate bare ground wire (just hot, neutral) you should be able to change your 2 prong outlets to 3 pround polorized ones and ground it by taking a short piece of solid copper ground wire and jump from the green ground screw to the neutral screw (silver) or into the back of the neutral screw if available. After all a proper ground wire goes to the same grounding/neutral bar in the main box.

As I said it would not be up to code but would this grounding serve the purpose. When you use a tester it shows the outlet properly grounded and not open ground.

Advise

You are 100% incorrect here. The fact that they terminate in the same place is completely irrelevant. Although is a common misconception, it can be a VERY dangerous one.

A neutral carries current. A ground does not and should not.
If you jump ground to neutral this brings the bare parts of the circuit to the same voltage potential as the neutral. Meaning, if you lose a neutral ahead of that receptacle the circuit will be looking for a way to find it's way back to the source. The source is the utility transformer. Since this and most other parts of the service are grounded to earth this is one path it will seek. Problem is the earth is a poor conductor, so you will get a shock, even a fatal one, but the earth, or other paths, will not be able to carry enough current to trip a breaker or blow a fuse.

PLEASE, you or others reading this or thinking about this, DO NOT DO THIS!!

Your allowed to replace 2-prong outlets with a properly labeled GFCI (Labels come in the package).

If you had a properly connected GFCI and you shorted neutral to ground UPSTREAM, it would trip.

Ground is a reference and should not carry current unless it's a fault. It' the zero point for your CATV, your stereo, your plumbing fixtures etc. Under fault and normal conditions in YOUR PROPOSED SCENEREO, the grounds will be at different potentials.

Your allowed to replace 2-prong outlets with a properly labeled GFCI (Labels come in the package).

If you had a properly connected GFCI and you shorted neutral to ground UPSTREAM, it would trip.

Ground is a reference and should not carry current unless it's a fault. It' the zero point for your CATV, your stereo, your plumbing fixtures etc. Under fault and normal conditions in YOUR PROPOSED SCENEREO, the grounds will be at different potentials.

Why does the plug in GAF-500 show proper ground when you hook up as I suggested? Before I did it the tester showed OPEN GROUND.

I'll urge you to read here: https://www.askmehelpdesk.com/electrical-lighting/grounding-bonding-surge-supprression-386501.html

Just lost what I wrote.

This tester: http://www.aemc.com/products/pdf/2117.60.pdf is used to probe ground currents. To be a ground, there must be little or no currents on the ground.

If lightning struck your panel ground, all ground potentials in your house would go up. Lot's of devices would be destroyed.

Why does the plug in GAF-500 show proper ground when you hook up as I suggested? Before I did it the tester showed OPEN GROUND.
Because the tester is being FOOLED into thinking there is a valid ground.