It's most likely not a setic tank issue, as you say its in an apartment building. If there IS a septic tank, it is nost likely full, and because they are not cheap to dump, the landlord may not have the money. Bug him if there is a septic tank.
If its on sewer, it could be clooged drains and P traps, which once the water slowly flushes down past the partial blockage, is releasing stench up through the toilet drain. If they have roto-rooter where ever you are, that is the answer, if not, you may have to remove the bowl, get a glove , spoon, stick, whatever it takes and manually clean all that disgusting gunk out. You will puke because if you think the thought of it is gross, having your fingers in it and the stench will overcome your senses.
Anyway, using a toilet brush or any brush (Don't let it drop in the hole) Scrub the hell out of it with straight javex. Getting down as far as you possible can, and until the brush comes up clean every time. Then you can replace the toilet bowl, by first scraping off the wax ring that was there (its a seal) and buying and placing a new one in its place. ( They are cheap about bucks at any plumbing store) This itself might be the problem, as a wax seal that isn't sealing will also release stink into the room.
Re faste your toilet, turn back on the water, and you should be good to go. A littler trick you can use is once a month drop a half a cup of 0-0-0- fertilizer down the toilet. (it's cheap too, like 7 bucks for a 25 lb bag) (it (the nitrogen)creates a chemical reaction which eats up the crap with an enzyme action) Its used for maintainng septic tanks (non-concrete ones) . It's not that environmentally friendly I guess, but it helps your pipes stay cleaner lots longer so you don't have to do that disgusting job again too soon. Hope it helps.