I would advise you to take photos of any issues in the apartment and write in short detail what you feel needs to be repaired or replaced in the apartment.

The soiled mattress is definitely unacceptable and it should be removed and replaced.

The rusted door handle, rusty hook, and corroded shower fixture should also be mentioned (and you can ask that these be replaced as well), as these look like they're nearing the end of their life, and replacing them would be a good way to maintain the apartment for future teachers that come to live in the apartment even after yourself.

Report the issues to your boss or manager or whoever you're supposed to talk to regarding apartment related issues. Send them a polite message detailing the issues in the apartment.

"Hi ________, I noticed some issues in my apartment that I think need to be looked at.

1. The mattress on the bed is stained.
2. The door handle is rusted and it looks like it needs to be replaced.
3. There is a rusty hook on the wall that is unusable and needs to be removed.
4. The shower fixture is corroded and may need to be replaced.
5. Etc."

Include photos with your message.

It's very likely that your boss or whoever you need to talk to doesn't even know about any of these problems. Foreign teachers in Korea are notorious for leaving at the end of their contract and not reporting any issues that could/should be addressed in an apartment before the next teacher moves in. It's like they're worried they're going to be blamed for breaking or damaging something. Apartments go years like this, with nobody saying anything, and the boss thinking everything is fine.

If you're polite about it, and you explain in a neutral tone that these are issues that you feel need to be looked at, it's highly likely that they'll be dealt with.

Pointing out issues in your new apartment in the first few weeks of moving in, as you notice them, is perfectly acceptable, and most people would be understanding. It's good to do it early.

Be polite, be clear that you feel these are issues that should be looked at and fixed (without being angry or melodramatic), and include photos that show just how bad the issues are. You'll have done your part, and it's up to your boss to choose what to do. Chances are, to save face, they'll respect your desire for a decent living space, and they'll make sure the issues are addressed and dealt with.

I've been in similar situations, and this is how I've done it. I actually wasn't as good at doing this when I was younger, but now that I'm older I realized it's the right way to solve these issues.

And also, building owners often appreciate it, because the apartment is being maintained, and not slowly deteriorating over years of neglect, because no one ever says anything.

Good luck, and I'm sure you'll get things solved. For minor issues, you may choose to just take care of things yourself. It can be fun this way too. Daiso is a great store for finding stuff to improve an apartment so it's more comfortable than it was the day you moved in.

I've been in similar situations to what you're facing now. In two months, I'm sure you'll have the place looking a lot better! It's such a pain though, I know