You can’t do it with the cable- The only way Palm indicates it can be done is with a clunky cell phone connection.
The only efficient way you could possibly do it is with a wifi SD card and even then the E’s processor, a TI OMAP 311 126MHz, is not that powerful so it’s not optimal for surfing. Palm doesn’t even show wifi SD card use as an option on the E so whether it will work even with the card is open to question. The E is not really built to surf.
One of the higher end Tungstens with integrated browsing and wifi like the W is really what you need. You can probably get a deal on a W as I don’t think they are selling that well, and Palm offered a half price deal on some W refurbs the other day for around $ 200 or so. .
It is possible to connect to the internet while hotsyncing. Both ePocrates and Taracon Pharmacopeia update their databases through an internet connection made with hotsyncing. I don’t know if this would work for you, but you could look into how they do it.