Could You Select All Layers Sequentially

Your first screen shot showed that you wanted the artboard numbered top to bottom then left to right, which the previous script that I posted did. You're second screen shot shows left to right then top to bottom. So this script does that:

#target photoshop
var doc = activeDocument;
var artArray = [];
var count = 0;
for(var i=0;i<doc.layers.length;i++){
    doc.activeLayer = doc.layers[i];
    if(isArtBoard ()){
        var dimArt = getArtboardDimensions ()
        artArray.push([dimArt[0],dimArt[1],doc.activeLayer])
        count++
        }
    }

artArray.sort(function(a,b){return (a[1]-b[1]) || (a[0]-b[0])});

for (i=0;i<artArray.length;i++){
    artArray[i][2].name = 'Artboard ' + (i+1);
    }

function isArtBoard()
{
    var ref = new ActionReference();
    ref.putEnumerated(charIDToTypeID("Lyr "), charIDToTypeID("Ordn"), charIDToTypeID("Trgt"));
    return executeActionGet(ref).getBoolean(stringIDToTypeID("artboardEnabled"));
}; // end of isArtBoard()

function getArtboardDimensions()
{
    var ref = new ActionReference();
    ref.putEnumerated(charIDToTypeID("Lyr "), charIDToTypeID("Ordn"), charIDToTypeID("Trgt"));
    var desc = executeActionGet(ref).getObjectValue(stringIDToTypeID("artboard")).getObjectValue(stringIDToTypeID("artboardRect"));
    var left = desc.getDouble(stringIDToTypeID("left"));
    var top = desc.getDouble(stringIDToTypeID("top"));
    return [left, top]
}; // end of getArtboardDimensions()