You can use Wine’s appDB to check whether an application will work with Wine. According to the OneNote page, some people have gotten the 2007 version working with a relatively recent version of Wine (which you should have if you have a recent version of Ubuntu). On that page you can also see what others have reported works and doesn’t work – though the reports aren’t that recent. If you try it, consider adding a report yourself :)
Running applications with Wine is a matter of right-clicking a .exe-file, and then select Open with.. -> Wine Windows Emulator (or something like that). Of course, you first need to install Wine through the Software Center.
That said, if you truly want stability, you can try running Windows inside Ubuntu using a virtual PC, through VirtualBox (which you can just install from the Software Center). After you install that, you can run it to install Windows inside it.
Also, I found a tutorial on running OneNote using Bordeaux, which apparently is a proprietary application based on Wine. Not sure whether it’s cool, but you can consider it.