Alice, How do I fix the leaky faucet that my landlord is too lazy to fix?
It could be that your landlord isn't legally obligated to fix it, since a leaky faucet doesn't render your apartment uninhabitable — although it might help to write him an official repair request in a letter.
If that doesn't work, you could always call a plumber and shoulder the bill yourself, but — even more fun (in my opinion) and less expensive: It's likely you can fix the leaking faucet yourself, and feel good about it to boot (or at least get more familiar with your sink's plumbing): This Old House has a useful guide for how to fix dripping faucets, and I found this two-part YouTube tutorial to be useful, too (and soothing). WikiHow has another. And you probably have a lot of friends who've gone through something similar and would love to show off their newfound (or old-found) plumbing familiarity if you asked them.
In the meantime, and especially if the noise is driving you to distraction, there's always the old tie-a-string-on-it method.