Could I use the parts from that car?
Probably not. In any case, all RC boxes work the about the same differing in # of channels, range etc.
You should start with something inexpensive like this kit. I am not recommending this model just using it as an example.http://www.hobbyking.com/hobbyking/store/__16239__HobbyKing_HK6S_2_4Ghz_FHSS_6Ch_Tx_Rx_Mode_2_.html
The little black box receiver 6 outputs must then be read and interpreted by the Arduino. This only makes sense if you intend to move toward semi-autonomous or fully autonomous operation. Otherwise the Arduino is pointless and direct RC control easier.
If you search this forum you will find others who have done similar projects. I'm no expert on RC systems.