Plumbing – Flushing upstairs clears a stopped toilet downstairs

The drain line is probably filled with hair and soap, chemical drain cleaners at best push the clog further down and make things worse in my opinion (now toilet backing up) with the exception of “mule kick” that will eat metal pipes and I am not sure if it is even available any longer. What you need is a real snake a small one to be run down the shower, and possibly a larger power one to be run down the toilet. Without cleaning the clog this will get worse this kind of snake you hold in one hand and spin pushing it down the drain a little at a time probably less than 2 or 3 chemical treatments. You should have the land lord take care of this incase any damaged pipe or other problems come up and you are not taking the $ hit for something that should be fixed by them.

Tyson's response in comment is spot on. Without knowing your house layout, you can follow the drain from the downstairs toilet to where it meets the pipe for the drain from the upstairs toilet.

Once you identify the drains intersection, look for another drain from any sink anywhere in the home that is PAST the intersection of the two toilets pipes AND closer to the main sewer drain.

If there is a sink, run water in it full, drain it, time it. Do it again, but this time flush the toilet. See if the water drains slow. This is just a simple test if do-able to "guess" whether there is a plug/buildup in the pipe where the toilets meet. Problem areas often will be at bends, turns, or where pipes are joined together. If there is buildup on the outside of the pipe, it could be a sign of buildup inside too.

There is even a possibility your issue is tree roots or broken sewer out to the main - it can be a pricey job and will only get worse.

If you can snake it, and are alright with opening the clean outs, go for it. Otherwise call a plumber. And be aware - if you were to snake haphazardly, you can do more damage than may already exist, especially if the pipes are old/brittle.