We own our house, so I have to handle this myself. Our problem is just like yours, maybe once a year. The source most certainly in our case is large trees in the front yard over the main sewer line. Apparently there is a cracked pipe and tree roots enter there and build up. Ultimate solution would be to kill the tree, then dig down and replace the cracked pipe - big money, and we like that tree!
When the main sewer backs up I rent a snake from a local equipment rental firm. Usually I get a 1/2" snake, 50 feet long, with an end piece that has sharpened edges on the arms to cut through light roots. I have used a motor-powered snake, but found it difficult to control when it jams. So I prefer the manually-cranked kind, even though using it gives me a real good workout cranking the thing. From the resistance to turning and traveling through the drain pipe I can tell when it has hit the tree roots. That is confirmed when I pull it out and it brings mashed and cut rootlets out with it.
Any house should have a cleanout access point on the main sewer line, usually at the base of the 4" line that goes down through the basement floor. You unscrew the cap there (may take a big wrench if it's old and seized), push the end of the snake down into that a bit, and start cranking. The snake will pull itself into the drain line because it is a spiral coil. Wear good leather work gloves and guide the snake so it does not fly around on you. From time to time you may want to pull the snake back out a bit and re-run through an area you think has shown resistance, indicating a clog. In fact, where there is such a clog, deliberately hold back on the snake so that it proceeds slowly through that area and breaks it up thoroughly. Once you've got some unclogging done, use a hose to run water down the line. Be careful at first - if you only got part of it unclogged, the line may still back up quickly! But if it flows clean and does not back up, then push a lot of water through to flush away the material you broke up.
When pulling the snake back out, sometimes you have to turn the snake backwards to get it unstuck, but generally don't do that. Turn the snake the normal way and use your gloved hands to pull it back out of the hole. Occasionally you get a snake with the end fittings screwed on, and turning the thing backwards just might unscrew that, leaving the end tool stuck in the drain pipe!
OP, obviously you won't be doing this - that is your landlord's responsibility, and it appears he / she is going to have it done professionally. Just wanted you to know your symptoms definitely sound like a plug (or more than one) in the main 4" sewer line from the house to the street main line, and a plumber certainly should be able to fix.
I also echo the advice to avoid chemical drain cleaners. They are useful, but only for certain types of clogs right under the sink. The original products like "Drano" were caustic crystals (strong bases). When placed in a drain with water they dissolve, releasing a lot of heat. The resulting hot caustic solution could attack typical fat and oil deposits that were the "glue" holding hair and stuff together, thus breaking up the clog. After some time for this process (30 minutes, maybe) you could use water to flush the broken-up clog all the way down, or maybe even use some plunger action to help. Later we saw on the market liquid caustics that do a similar job, but lack the heat release property. Then came acid liquids. Acids can do a similar attack on fats and oils, sometimes not quite as well, and they have the downside that they attack metal pipes whereas caustics do not. But with plastic plumbing throughout a newer house, that is not a big issue. The newest products use enzymes for the attacks on fats and oils. The are slower, perhaps, but the are not nearly so hazardous as strong caustics or acids. But in ALL of these cases. the key is that the chemical needs to be kept in one location and concentrated for at least a half hour to do the job. So they are ideal for a clogged trap under a sink, but absolutely useless in a long horizontal drain line. And you certainly do not want to try to put enough into a sewer system to push it all the way down to a clog half way to the roadside to attempt attack on a clog there. Besides, that clog may not be hair stick together with grease, so the chemicals may not be able to break it up, anyway!