Well is the system 64-bit or not (I don't see it in your post, might be because I don't know exactly what kind of processor E6600 is)? If it is not 64-bit, you obviously can't install 64-bit system. If it is, you can install it if you like.
Before buying make sure the hardware works with FC6 (or some other Linux as well); ASUS boards have caused some headache for some users, so if you can't do anything else (nor find information from the web to ensure it works), try making a deal that you can get your money back if the machine doesn't work.
EDIT: there are pros and cons using 64-bit system or not, the first thing that comes into my mind is that some apps that run on 32-bit system refuse to run in 64-bit (Wine is one, but it can be made to run on 64-bit also). Anyway the amount of such programs is (decreasingly) small, so if you have a possibility of installing 64-bit OS then do it; otherwise you're just wasting some machine power