There are many ways to connect a RO unit. Some of the more common are faucet adapters that screw on, saddle valves for piercing into a supply pipe (my least favorite way to connect), or hose end adapters (my preferred way to connect). I hooked ours up to the supply line for our washing machine. I bought a Y-adapter to convert the single hose connection into 2 connections, then used a hose end adapter to connect our RO unit. The waste line from the RO unit just goes down the drain intended for the washing machine.
In all reality, connecting a RO unit really doesn't require messing around with under the sink connections unless you need to go that route. Some of the other options, if convenient based upon the set-up of the apartment, require no tampering with the existing plumbing (which is a good way to make sure you get your security deposit back).