Can I control my Arduino using just a 2.4GHz Wireless Transceiver and a wireless Xbox 360 controller?
Short answer is no, unless you have the machine and tools to do reverse engineering on the xbox controller proprietary 2.4Ghz protocol and you must have xbox 360 itself to see what goes on between Xbox 360 and the xBox controller. The time and money spent on these tools(hardware and software) aren't worth it. 

The easiest way to do this is to buy the xbox controller receiver, Arduino USB shield and you can easily use your xbox 360 controller with your Arduino.