Unfortunately, there is no browser that will give you full access OWA 2003′s feature set. Microsoft incorporated a lot of browser calls that only work in IE6 or above. No browser on the Mac can support these calls. If you HAD to pick a browser to do it on, Safari would probably be the best but not much better than Firefox. And, as sad as it is to say, IE is probably the worst since it’s been abandoned.
The Entourage answer will work to a degree and will give you access to the folders, although I’m still not sure it will give you everything you are looking for.
Your easiest solution is to use Entourage 2004 to connect over http:// to the exchange server. By entering your Outlook Web Access URL into Entourages account information field, you can use Entourage wherever you can go on the internet. Even SSL will work.
You can also install iSync and synchronize Apple Mail with the Exchange Server.