Air conditioning, or cooling of the sort you are describing, uses a compressor to compress the gas. In so doing, it gains thermal energy (heat) which has to be removed from the system via a heat exchanger. This means that your description of a compressor or a heat exchanger is close, but not close enough. You have to have both.
The heat of compression is removed by a heat exchanger. The most common design uses forced air to cool the compressed gas. Heat is removed from the gas, the air around the heat exchanger absorbs it, so to speak.
In the process of exchanging heat to the environment, the gas condenses and becomes a liquid (or a more dense gas) and is passed on to the evaporator, which is the cooler part of the system. A pressure reducing device is also incorporated into the system.
The liquid evaporates (hence the name), absorbing thermal energy from the environment, cooling it. In the process it becomes a gas again and the cycle continues.
The reference to figure three of your link points to a liquid to gaseous nitrogen heat exchanger. This is a net-loss cooling system. The LIN evaporates and cools the intermediate heat exchanger in a manner similar to the above description, but the nitrogen is lost to the atmosphere. You'd have to have a recovery system in place to capture and re-compress/re-condense the nitrogen. That stuff is pretty expensive compared to a conventional refrigeration system and conventional refrigerant.
Your last question regarding the amount of LN2 would have to have a specific duration, as the nitrogen isn't going to stick around for long and the heat would return as soon as the supply tank was empty.
Consider this experiment. Ordinary rubbing alcohol (or any alcohol!) will evaporate, creating a cooling action. While the bottle is full and dripping onto a container, the container is causing the liquid to evaporate (just as nitrogen would). Eventually, the container would reach the evaporation temperature of the alcohol and it would not get cooler. That temperature is higher than nitrogen's, but the concept is valid.
After the alcohol bottle is empty, the container that was cooling will begin to warm again. If the entire assembly was encased in another container and the evaporated alcohol was recovered, compressed and cooled, it would work longer, but you're back to the original refrigerator and compressor system.
A less complex example is a typical shop air compressor. As the compressor is running, the storage tank will become warm. The heat exchange portion is the tank mass and airflow around it. Eventually the tank temperature becomes that of the surrounding air.
When the air is released, the reduction of pressure results in cooling and can be felt on the tank.
It is simpler and less expensive to stick with conventional refrigeration unless you need unusually low temperatures. Also consider to research ammonia cooling systems, used for some winter sport venues such as bobsled and luge. (can be dangerous)
