Intel® Core™2 Duo Processor E4300 - Is it a 64 Bit Processor?

Yes, all Core 2 Duo chips are 64-bit capable, that is includes the EM64T feature. Of course, if your running a 32-bit OS then it'll be no different then a 32-bit chip, but if you put a 64-bit OS on your system the Core 2 Duo chips will handle it properly. So yes it is capable of running 64-bit Windows.

however please note;

Intel® EM64T requires a computer system with a processor, chipset, BIOS, operating system, device drivers and applications enabled for Intel EM64T. Processor will not operate (including 32-bit operation) without an Intel EM64T-enabled BIOS. Performance will vary depending on your hardware and software configurations.

http://processorfinder.intel.com/details.aspx?sSpec=SL9TB

According to Intel, yes, it is a 64 bit processor.