The drain is the problem, find where the hose enters the drain, whether it enters it's own drain pipe or the side of the disposal or an airgap that is connected to the top of he sink.  Disconnect the hose clean both internal areas, hose end and where ever the connection is inside that drain as well, even if you use a flat head screw driver clean it really well.  reconnect it and run it thru a drain cycle a couple of times this should rinse out the remainder of what ever was plugging the drain.  If the air gap is completely clogged the water when the D/W drains, water will come out, if it is a air gap that protrudes thru the wet side of the sink, be sure the air gap is cleaned on both sides.   I totally agree with Bob with it being a rental, call the landlord, let him do it.