How can I temporarily fix a cracked bathroom sink?
You can probably fill the crack with epoxy to stop the leak.  It might not look great but it will do the job.

Make sure the area is clean and dry, and that you follow the directions on the packaging.  If there is any chipping make sure to brush away any loose debris so that the epoxy will bond correctly.

As far as what type of epoxy to use, Home Depot carries a Marine Epoxy which they describe as:


  Marine Epoxy is a permanent, waterproof adhesive. The 2-part formula is ideal for projects requiring high bonding strength on surfaces exposed to water immersion. The adhesive can be applied and will cure underwater. It can be used on glass, metal, wood and other common materials.


This sounds like it would work well for your application.