Given that (according to your link) it uses a regular brushed-DC motor and ESC, it likely also uses a fairly standard RC hobby servo for the steering control, and likely a standard 2.4 GHz RC radio system.
In short - you should be able to control this with the Arduino, using the Servo library.
You could do this one of two ways:
1) Replace the RC receiver with the Arduino
2) Augment the RC receiver with the Arduino
The second option opens up a lot more possibilities than the first, but is more difficult to implement; basically, you take the outputs of the receiver, and input them into the Arduino - which, using software, interprets them, then outputs signals to the motor controller (ESC) and the steering servo.
This could be a simple "pass-thru" system (like the Arduino wasn't even there), or you could do such things as "macro controls" (replay back previous recorded "moves" - on command from the receiver), as well as take input from other sensors (like an ultrasonic distance sensor, for instance) and "takeover" or "augment" the input to the car (for instance - you direct the car toward an obstacle, and the car could auto-magically slow down, veer away, reverse, etc - to keep you from crashing the car).
While more difficult to implement (you basically need to come up with a non-blocking method to read the RC receiver PPM outputs while simultaneously sending those outputs to the servo and ESC) - as you can see, it opens up more possibilities than a straight replacement. Fortunately - if you do some digging, you can find various libraries and such which can help with the implementation and control.