There are three things that need to all be present for the full effects of a 64-bit platform to be realized:
1: A 64-bit CPU (which you have)
2: A 64-bit OS (which XP is not)
3: A 64-bit program
You can get away with just #1 & #2, though it's not necessarily a given that an OS will have 32-bit backwards compatibility. It's something companies choose to add in.
So if you wanted to upgrade to Vista or later, there are officially supported 64-bit versions of Windows. While there was a 64-bit version of XP, it should be avoided. Driver support was always poor at best, and Microsoft ceased all support for it several years ago. Support for it's 32-bit sibling is drawing rapidly to a close, and so the wise person would consider XP a non-option for running on any computer that will be connected to the Internet.