The CPU is fine for a 64 bit OS:
The rest of your PC is probably Ok, but you should check Microsoft's website to see their requirements.
Well, Intel does not list an E6500R on their web site, but E6500 is listed, check the link below:
The E6500 is a 64bit CPU, and the differences between the E6500K are minimal and insignificant regarding the OS. The MS Windows Advisory may be recommending the 32bit OS due some 32bit software and/or drivers installed on your PC now. Your mother boards G41 chipset will support CPUs up to the Core 2 Quad 9000 series, so the board itself is not the limitation. Do you have 1 GB or less of memory, that might limit you to a 32 bit OS.
Besides having 32bit drivers and software, or a small amount of memory, I don't understand why the Advisory program recommended a 32bit OS.