Yep. When you have a plumbing question, just ask a FatBaldGuy. (kidding! I’m kidding!!)
Like he says, Sounds like you’ve got a moderate plug in the stack just below your sink. Your toilet and shower would join that stack slightly below where your sink joins, so there’s a possibility of a plug there. As FBG says, if it’s a slow drain, your neighbor’s sink doesn’t trigger the backup, but the large amounts of water from a shower will.
Is it a duplex kind of apartment, or are the units stacked, with identical layouts above each other like more modern buildings? If it’s an old building with unique floor plans, your neighbor’s fixtures may not align directly with yours, i.e. his shower may join the stack that your sink is on, but the toilet goes to a different stack altogether. In a more modern building, the fixtures all join a common stack.
But you’re right, as the lowest tenant, you are closest to the front lines. In other words, when there’s a plug or probelm, you’re the first to know. So actually, your landlord should be happy to hear from you before the problem gets really bad (it’s a good theory anyway)
Somewhere below your apartment, all the vertical stacks are joined together and take a sharp corner to be joined with the stately river of human offal that is searching for a treatment plant, like salmon search for spawning grounds (except sludge flows downstream) this stream is called ‘black water’.
Since your sink also occasionally backs up, it sounds like there may be a slowage in lateral between the building and the City sewer as well. A properly designed system shouldn’t back up except in extreme conditions, like flooding.
The plumber really needs to check out the building lateral also. Older buildings have a tendency for the lateral to slowly become more constrained, until things like toilet paper, feminine products, and Mr. Hanky become stopped, then the whole system goes kerflooey. There’s an episode of Dirty Jobs where they go in to deal with one of these.