It's not uncommon in older buildings with old wiring to have some or all outlets ungrounded. Ground wires didn't become required until sometime in the 1960s or 1970s.
Ideally, they should be protected by a GFCI - either an outlet with the TEST and RESET buttons on it (which may daisy chain to other outlets, so the buttons are not necessarily at the same spot as you are plugging in), or a GFCI breaker in the electrical panel.
Without a ground wire, surge protectors won't work, so your UPS or power bar with surge protection won't protect anything if there's a nearby lightning strike or anything like that. So equipment could be damaged.
If it's without GFCI, you're also missing the protection from electrical shocks if a loose wire electrifies metal parts of things you can touch - the computer case, the metal base of a lamp, etc. Usually that just results in a painful shock, but if all the right conditions happen it can cause life threatening injuries.
Getting the landlord to update the wiring would be the best solution, but that's likely a very expensive solution that most wouldn't do unless forced to by code. And they likely aren't - old installations are usually grandfathered in so you can keep things running the way they were until you make any major changes or upgrades.
If there are no GFCIs, having the landlord install those would be the next best solution, is relatively cheap, and at least offers protection from shocks - though still no surge protection to save your equipment.
If your landlord is completely unwilling to do anything, you can still use it. It will probably be ok. All of these protections you're missing help make sure things are safer if something else goes wrong, but under normal operating conditions don't make any difference.
You can, if you are still worried, get portable GFCI devices that add that ground fault protection to an outlet. They look like little extension cords with test and reset buttons on one end. Like these: https://www.homedepot.ca/product/husky-portable-gfci/1000658960 - anything you plug into that will be GFCI protected so that it is much less likely for you to get a fatal shock if a metal case becomes energized. Still doesn't make surge protectors work though.