Electrical outlet right above sink
This might possibly have been legal in your jurisdiction around the time Franklin tied a string to a kite. As you've figured out, it's not prudent.

I know you said you've ruled out the possibility of a GFCI being upstream, but have you tested with one of the handheld interrupters? Reason I ask is that GFCIs sometimes turn up in the strangest places. 

The easiest fix I can think of is to replace it with a GFCI (using one of the hot legs and capping/continuing the other hot within the junction box).

I suspect your landlord is responsible for fixing this, but you'd have to check with your local powers-that-be. You might get points for fixing it yourself and charging the landlord for parts and a bit of labour.