You're stuck.
My brother does construction/remodeling in Oregon/Washington.
The Apartment owners/manager are not required to upgrade those outlets if it is an older homeapartment.
However, if the apartment were to catch fire, and then apartment rebuilt/remodeled, then they would have to conform to the new codes, as most likely the fire damage would require them to strip the apartment down to sticks and rebuild, electric and plumbing and all.
My brother just went through this problem with a house in Vancouver, where a meth lab caught the house on fire. He's fighting with insurance company because all the electrical work they are not paying for, as the homeowner did NOT have upgrades for code compliance in the insurance policy, but the city required him to upgrade the home to conform with the modern codes for compliance with city regulations in order to pass inspection.
A GFCI Outlet that is NOT properly grounded, won't help much either, but it will be better then no GFCI.
The only thing you can do is to bury a connected rod into the ground, and also then run a wire and connect it to the city water pipe that runs into the apartment, then you should be protected. It's pretty easy to do, but will most likely require you to get underneath the apartment to find the inlet for the city water.
Keep in mind though, if you get caught by a city inspector while doing this, and the property is NOT owned by you, they will fine you, and it is NOT cheap. I remember my brother was fined on a job installing a new wall plug, (he subs out the electrical and plumbing usually) and the inspector fined him $1500. The inspector made a surprise visit and kinda let himself in to inspect and saw my brother screwing on the wall plate. My brother is a licensed contractor, but NOT a licensed electrician.
At my house, we recently updated our meter to a 200A panel from a 60A. The electrician had to bury a rod and connect to the city water pipe as well. The whole job, with a new meter, cable going to county electric connection, new panel box, and the buried rod, was $1700.00. I then installed a dedicated 3-prong grounded outlet, 1 for each bedroom. We connect our computers/TV's to these outlets. The rest of the outlets are still the 1950's 2-prong outlets.