I just moved into an apartment with a side hose attachment on the kitchen sink. With the water going full blast, I hit the hose lever and hardly any water comes out, barely a dribble. However, it does completely cut off the water from coming out of the faucet, which leads me to believe it is not a problem with the diverter. Also strange is that when the water is cold, slightly more comes out of the hose than when it's hot.
Most likely the sprayer head is clogged with mineral buildup.
Soak it overnight in vinegar and see if that doesn't open it up.
If it doesn't, just ask the apt landlord to replace it. They're inexpensive and easy to change.