Google-sheets – Copy from individual cell in Google Sheets Table Chart

I don't think this is possible, since there is no onClick-trigger.
There are alternatives, though, using scripts. I've built two different ways.

Dropdown
Use a dropdown where you select the number of the column to choose from.

A button (drawing)
Click on a drawing with a script assigned to it updating the scripts source range.

See example of both here. The "Click me" drawing has the script updateChart-function assigned to it.

Since you have your chart-ranges on the x-axis I've used Mogsdad's method to Switch rows / columns programmatically. The method requires the 2D Arrays Library, read here how to install it. It also requires you to have a sheet named Scratch, you can hide it, though.

Here's the code used:

function onEdit(e) {
  var sheet = SpreadsheetApp.getActiveSheet();
  if (sheet.getName() == "Chart Sheet"){
    //var sourceRange = e.source.getActiveRange();
    var sourceRow = 1;//sourceRange.getRow();
    var sourceColumn = 1;//sourceRange.getColumn();
    var sourceRange = sheet.getRange(sourceRow, sourceColumn)
    var sourceValue = sourceRange.getValue();
    var lastCol = sheet.getLastColumn();
    var newRange = sheet.getRange(sourceValue + 1, 2, 1, lastCol);


    if(sourceRow == 1 && sourceColumn == 1){ //Address to cell with dropdown, placed in corner not to interfere with getLastColumn

      var srcData = sheet.getRange(sourceValue + 1, 2, 1, lastCol).getValues();

      // Transpose the table (using 2D Array Library)
      var scratchData = ArrayLib.transpose(srcData);

      var numRows = scratchData.length;
      var numCols = scratchData[0].length; // assume all rows are same width

      // Write scratch values to scratch sheet.
      var scratchSheet = SpreadsheetApp.getActiveSpreadsheet().getSheetByName("Scratch");
      scratchSheet.getRange(1, 1, numRows, numCols).setValues( scratchData );
      SpreadsheetApp.flush();



      var chart = sheet.getCharts()[0];
        var oldRange = chart.getRanges()[0];
        chart = chart.modify()
        .removeRange(oldRange)
        .addRange(scratchSheet.getDataRange())
        .build();
        sheet.updateChart(chart);
      }
    }
}

function updateChart(){
  var ss = SpreadsheetApp.getActiveSpreadsheet();
  var sheet = ss.getActiveSheet();
  var lastCol = sheet.getLastColumn();
  var row = sheet.getActiveRange().getRow();
  var newRange = sheet.getRange(row, 1, 1, 3);

  var srcData = sheet.getRange(row, 2, 1, lastCol).getValues();

  // Transpose the table (using 2D Array Library)
  var scratchData = ArrayLib.transpose(srcData);

  var numRows = scratchData.length;
  var numCols = scratchData[0].length; // assume all rows are same width

  // Write scratch values to scratch sheet.
  var scratchSheet = SpreadsheetApp.getActiveSpreadsheet().getSheetByName("Scratch");
  scratchSheet.getRange(1, 1, numRows, numCols).setValues( scratchData );
  SpreadsheetApp.flush();

  var chart = sheet.getCharts()[0];
  var oldRange = chart.getRanges()[0];
  chart = chart.modify()
  .removeRange(oldRange)
  .addRange(scratchSheet.getDataRange())
  .build();
  sheet.updateChart(chart);
  sheet.getRange(1, 1).setValue(row - 1); //This row is only necessary if both methods are used
}

I hope this helps, ask me if you have any questions!

If the row you are using for labels contains dates, the spreadsheet supposes that you need an accurate representation of time. So it arranges the data chronologically and marks the time axis at equal time intervals:

If you want the specific dates to be used as labels, this means they should be used like text labels. Select the top row and apply Format->Number->Plain Text:

Now the columns are labeled with the text from the top row, and they are presented in the order of appearance in the spreadsheet.