3-prong outlets without ground are legal if they are GFCI protected, with "GFCI protected" and "No equipment ground" stickers.
GFCI provides a "zone of protection". Let me explain.
Electricity moves on wires. It flows in loops so there must be two wires - out and back again. (ground is not used normally).
A GFCI module monitors the two normal wires. It measures all the power "going out" and make sure all of it "comes back again". Like a playground monitor making sure if 44 kids go to the playground, 44 kids come back.
A mis-count means electricity is going somewhere abnormal - possibly shocking someone or risking a fire. A GFCI will then "trip" or shut off the power.
So --- given the way GFCI works, it casts a "zone of protection" to all wiring downstream of it. (Any outlets whose two active wires are fed off the GFCI.)
This means the GFCI can be anywhere.
You know it's not at the breaker because there are no TEST buttons on them. However breakers with GFCI cost $40 while $16 buys either a dedicated GFCI module (aka deadface) or a GFCI+receptacle combo device (the normal socket with Test/Reset buttons).
The smart play for the landlord is to fit a $16 device at the first receptacle location on the circuit, apply the stickers and call it good.
So you really need to search the entire unit for any sockets or deadface sockets with the distinctive "Test" and "Reset" button. If pushing "test" on this device kills power to the sockets you are worried about, then it is protected.
A thing that zaps you has a problem
A piece of equipment that zaps you has a problem, regardless of whether it is grounded. Now it's possible the ground has been "saving you" but it shouldn't need to save you.
The simple fact is that this leakage which is zapping you is also going to trip a GFCI, and your device won't work and you'll have to fix it or replace it. So GFCI won't do what you want, it will only provide safety.
At the end of the day, the zappy thing needs to be fixed. If it's a PC, change the power supply, preferably with "not junk".