Replacing bathroom sink popup drain if sink has hairline cracks
I'm a bit confused by the depressed nature of those cracks, but I agree with UnhandledExcepSean. They're probably cosmetic. I've worked on some old sinks and have never had one disintegrate.

I'd look at the underside and see whether any of the cracks translate through. I'm guessing that if they did you'd have substantial leakage already.

Also, you could ask the property owners if they're willing to compensate you for expenses. You might get a new sink out of it.