The AMD Athlon II X2 250 processor is 64-bit, so you can run either the 32-bit (i386) or 64-bit (amd64) version of Ubuntu. You have enough RAM that you might well experience a noticeable improvement in performance even if you ran mostly 32-bit applications (so long as you multitask...which everybody does these days). And if you use the 64-bit version of Ubuntu then you'll be running mostly 64-bit applications, because by default you'll get 64-bit packages (so then you benefit from the CPU instructions that are present in the amd64 architecture but not in 32-bit architectures...or not in all 32-bit sub-architectures).
As of Ubuntu 11.10, multarch support is enabled by default, so you can install the 32-bit versions of applications or libraries if you need them. Therefore, even if you have 32-bit-only applications not provided by Ubuntu software sources (for example, a proprietary binary-only application that only provides a 32-bit version), it shouldn't be significantly more difficult (and would likely not be more difficult at all) to get that running smoothly on a 64-bit Ubuntu system than on a 32-bit Ubuntu system.
So 64-bit is probably the best choice for you, based on the information you have provided. But you're unlikely to go wrong either way...unless you have applications that benefit enormously from 64-bit support, like some scientific and computational applications.
Whether you install your Ubuntu system normally (which is usually the best choice) or with Wubi, you can use either the 32-bit or the 64-bit version of Ubuntu. Even if you're using a 32-bit version of Windows, your Wubi system can still be the 64-bit version of Ubuntu.
See the Ubuntu multarch specification page and the Debian multarch wiki page for more information on the multarch feature (enabled by default in Ubuntu as of version 11.10). See this listserv post and this news article for some more information about multarch in Ubuntu. See this question for general information about the differences between 32-bit and 64-bit Ubuntu systems (not specific to Ubuntu 11.10). See this question for information about why the Ubuntu website has referred to the 32-bit version as recommended.