作業中のjsFiddle  
$(document).ready(function() {
    chart = new Highcharts.Chart({
        chart: {
            renderTo: 'container',
            type: 'line'
        },
        title: {
            text: 'Monthly Average Temperature'
        },
        subtitle: {
            text: 'Source: WorldClimate.com'
        },
        xAxis: {
            categories: ['Jan', 'Feb', 'Mar', 'Apr', 'May', 'Jun',
                'Jul', 'Aug', 'Sep', 'Oct', 'Nov', 'Dec']
        },
        yAxis: {
            title: {
                text: 'Temperature'
            },
            plotLines: [{
                value: 0,
                width: 1,
                color: '#808080'
            }]
        },
        plotOptions: {
            series: {
                dataLabels: {
                    enabled: true
                },
                events: {
                    //Here I'm able to move the labels by using the click and shift-click. 
                    //I wish to do this by dragging
                    click: function(event) {
                        var current_y = event.point.dataLabel.attr('y');                        
                        if(event.shiftKey) {
                            var new_y = current_y + 2;
                            event.point.dataLabel.attr({y: new_y});
                        } else {
                            var new_y = current_y - 2;
                            event.point.dataLabel.attr({y: new_y});                            
                        }
                    }
                }                    
            }
        },
        series: [{
            name: 'Tokyo',
            data: [7.0, 6.9, 9.5, 14.5, 18.2, 21.5, 25.2, 26.5, 23.3, 18.3, 13.9, 9.6]
        }, {
            name: 'New York',
            data: [-0.2, 0.8, 5.7, 11.3, 17.0, 22.0, 24.8, 24.1, 20.1, 14.1, 8.6, 2.5]
        }, {
            name: 'Berlin',
            data: [-0.9, 0.6, 3.5, 8.4, 13.5, 17.0, 18.6, 17.9, 14.3, 9.0, 3.9, 1.0]
        }]
    });
    //Second attempt to grab a point. Works for chaning attributes
    var point = chart.series[0].data[0].dataLabel;
} , function(chart){
});
 $(function() {
   console.log(chart) ;
       $( "#draggable" ).draggable()
       .text(chart.series[0].data[0].y + 'drag and drop me     ');
     });
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