Roadsync provides Direct push capabilities with Windows Mobile 2003

Sync With Microsoft Exchange 2003 & Corporate Outlook Email Via Remote Email Access on Your Smartphone which doesnt have to be running Windows Mobile 5 (but it does need to be Windows Mobile 2003 - which still leaves my original axim out of the loop.)
It is not free (probably $99) but there is a 30 day trial so that you can check it works first.
It looks like it doesn't support notes either - something the native Exchange Direct Push doesn't do too.