Welcome to the world of HPC...
But as much as we love it, it is an abandoned platform
There is no IE upgrade short of a complete OS upgrade, as is the case with all CE devices. And there's no OS upgrade for Jornada's.
There are alternate browsers, such as NetFront, written for PocketPC. But in order to get it work on an HPC, you need some controversial "hecking". If interested, I'm sure there are a couple of threads about it.
As for specific webmail sites on IE4... they all depend. I recommend starting another thread to ask for help on that.
If your company has POP3 mail, use Inbox, as it works for most servers. It even accepts attachments