Preload content in StageWebView
I don't think there's anything that can be improved to your approach.

Minor improvements do apply;

function showWeb(e:Event=null){
webView.viewPort = new Rectangle( 0, 0, stage.stageWidth, stage.stageHeight);
}


could use some indentation,

btn.addEventListener(MouseEvent.CLICK,  showWeb)


contains a double space and is missing a semicolon.

But I don't think it's possible to preload the StageWebView via some function.

What you DO want to take care of is any errors that might be thrown your way.

For that, add a listener: (code from this SO answer)

webView.addEventListener(ErrorEvent.ERROR, onError);
function onError(e:ErrorEvent):void 
{
    trace("Page is not available.");
}