What can I do about my kitchen sink that has clogged suddenly and completely?
Usually it is illegal for the landlord to charge for building repairs.  The only time a landlord should be able to charge is when the damage is not from normal wear and tear (like dumping a pot of rice down the garbage disposal).  

It's hard to guess what is causing the blockage but bet it is related to the Nu-Flo coating de-laminating and clogging the drain.  It is best to notify the landlord to have a plumber come out to auger the line.

The item on your sink is an 'air gap' to connect the dishwasher to the disposal.  The air gap prevents the garbage disposal from black flowing into the dishwasher.