How do I drill additional holes for adjustable-height shelving?
Do you need more shelves, or do you just need one of the existing shelves to be at a different height?

If it's the latter, it may be easier for you to add some supports underneath one of the existing shelves like this:



The supports should run the depth of the cabinet, and you can attach them with command strips or double-sided tape.  Be careful removing them because shelves are often covered with paper that's printed to look like wood.  

If you need more shelves, it may be better for you to build a new shelf using the same idea, but have the new shelf resting on one of the existing ones.  You'd lose a little bit of space on the lower shelf, but you'd avoid the risk of breaking your rental agreement and/or losing your deposit.  You'd also need to be careful not to overload the lower shelf or the whole thing could come down.