I promise, this issue has nothing to do with your bathroom cleaning habits. Caulk is not permanent; it degrades over time. As it degrades, more and more moisture sticks around and fosters mold, which stains it dark. Even cleaning your bathroom daily (lol who does that, not me) wouldn't prevent it. There's just going to be moisture in a shower. Especially if you're unlucky like us and have no fan or ventilation system in the bathroom.
I am also Not Handy, and I just replaced our caulk a few months ago because it was gross, stained, and peeling off, and our super/landlord ignored all our requests to fix it. It really wasn't that bad. It took a full day because of all the steps where you have to wait, but it wasn't that much actual work.
1) I found it useful to get a caulk softener. If you're worried about scratching the tub, this is a way to go. I put it on all the old caulk, left it for two hours, and it was much easier to scrape off the old caulk.
2) Once the old caulk was gone, I used a bleach mold/mildew spray and a toothbrush to thoroughly clean the edges and kill any remaining mold/mildew. Then I let it dry thoroughly, another few hours.
3) Then I went for it. There are a lot of youtube videos and howtos if you google the process. Just pick the one that seems most doable and give it a shot. I didn't do any testing on a cardboard box first, though that would have helped. My first attempt looked ridiculous, but I just wiped it off while it was still wet and tried again. Even if it dries messy, you could always scrape it off and redo it. The one tip I would say is to cut open the caulk gun with a much smaller opening than you'll think you need. It spread out a lot when I smoothed it with my finger.
It doesn't look quite the same as it would if a professional did it, but it's certainly an upgrade from the old caulk. It's not stained, it's not peeling, and it's actually moisture tight now.
You honestly can't screw it up. Even if there is a small gap somewhere, it's not going to let through enough water to get into your neighbor's apartment. Worst case, it would just let moisture stick around and foster mold - exactly the situation you're in now. You can do it!
If you're still nervous, you could try asking your landlord. He should know that it has nothing to do with cleaning habits. This is just a necessary part of bathroom maintenance sometimes. Just clean the bathroom right before he comes over to do it to make yourself feel more comfortable about it. (Or am I the only one who cleans for my super so I won't get judged....?)