You could get the values in your header row, and search for the required header in those values using indexOf().
function onEdit(event)
{
var timezone = "GMT-4";
var timestamp_format = "yyyy-MM-dd HH:mm:ss";
var sheet = event.source.getActiveSheet();
// note: actRng = the cell being updated
var actRng = event.source.getActiveRange();
var index = actRng.getRowIndex();
var headers = sheet.getRange(1, 1, 1, sheet.getLastColumn()).getValues();
var dateCol = headers[0].indexOf('Last Updated');
if (dateCol > -1 && index > 1) { // only timestamp if 'Last Updated' header exists, but not in the header row itself!
var cell = sheet.getRange(index, dateCol + 1);
var date = Utilities.formatDate(new Date(), timezone, timestamp_format);
cell.setValue(date);
}
}