IIRC, grounded outlets weren't required until 1958 in the US's NEC for residential construction (grounded to the metal box, not a 3 prong plug).
If it's feasible, you could see if there's access to a ground rod (or install one), run a dedicated ground between the box and ground rod, and replace the existing outlet with a GFCI. Done this a few times in older construction when there wasn't a separate ground present.
Given the situation, this tends to be an economical solution to retrofitting older wiring, so may be a viable solution for your landlord (electrician will likely discuss this option).