Hi,
Out of the box ArcGIS Explorer does not offer an export to PDF option. However, there are several free and paid-for PDF converters availabler which you can call programmatically within a custom Add-In (e.g. you might write an "Export to PDF" Button Add-In). I recommend typing, for example, "free PDF converter C#" into your favourite internet search engine - this will reveal some options that you could look into.
There are a number of ways to share content with ArcGIS Desktop 9.3.1, which are improved again at the imminent ArcGIS Explorer 1500 release - for example through the sharing of Layer Packages. I suggest watching out for this new release rather than invest too much time now in trying to find a way to share content between Explorer and Desktop.
Regards
Mike