How can I determine the rotate angle of an artLayer?

You can't get the angle directly from the layer's transparency. You can load the transparency as a selection. Make a path from that selection. Then examine the pathPoints to find an angle in the path. With a simple rectanglar shaped layer you should be able to get close to the correct angle if it was only rotated once and less than +- 89 degrees.

function loadTransparency(){
    var desc = new ActionDescriptor();
        var ref = new ActionReference();
        ref.putProperty( charIDToTypeID( "Chnl" ), charIDToTypeID( "fsel" ) );
    desc.putReference( charIDToTypeID( "null" ), ref );
        var ref1 = new ActionReference();
        ref1.putEnumerated( charIDToTypeID( "Chnl" ), charIDToTypeID( "Chnl" ), charIDToTypeID( "Trsp" ) );
    desc.putReference( charIDToTypeID( "T   " ), ref1 );
    executeAction( charIDToTypeID( "setd" ), desc, DialogModes.NO );
};
function makePathFromSelection() {
    var desc = new ActionDescriptor();
        var ref = new ActionReference();
        ref.putClass( charIDToTypeID('Path') );
    desc.putReference( charIDToTypeID('null'), ref );
        var channelRef = new ActionReference();
        channelRef.putProperty( charIDToTypeID('csel'), charIDToTypeID('fsel') );
    desc.putReference( charIDToTypeID('From'), channelRef );
    desc.putUnitDouble( charIDToTypeID('Tlrn'), charIDToTypeID('#Pxl'), 2.000000 );
    executeAction( charIDToTypeID('Mk  '), desc, DialogModes.NO );
};
function activePathIndex(){
    try{
        var ref = new ActionReference();
        ref.putEnumerated( charIDToTypeID("Dcmn"), charIDToTypeID("Ordn"), charIDToTypeID("Trgt") );
        var desc = executeActionGet( ref );
        return  desc.getInteger(charIDToTypeID("TrgP" ));
    }catch(e){}
};
function measureAngle(path){
   var res = new Array;
   var lineStart = path.subPathItems[0].pathPoints[0].anchor; 
   var lineEnd = path.subPathItems[0].pathPoints[1].anchor;
   var a = Math.max(lineStart[0],lineEnd[0])-Math.min(lineStart[0],lineEnd[0]);
   var o = Math.max(lineStart[1],lineEnd[1])-Math.min(lineStart[1],lineEnd[1]);
   var c = Math.round(Math.sqrt((a*a)+(o*o)));
   res.push(c);
   var ang = (180/Math.PI) * Math.atan2(o,a);
   if(lineStart[1] < lineEnd[1]){//negative angle 
      ang = -ang; 
   };
   res.push(ang);
   return res;
};
loadTransparency();
makePathFromSelection();
var tempPath = app.activeDocument.pathItems[activePathIndex()];
 var angle = measureAngle( tempPath )[1];
 tempPath.remove();
 alert( Math.abs(angle) );