Oh I found the solution, apparently I wasn’t too far away.
My where clause was faulty,
I have now changed my code as follows:
global $wpdb;
$foobarOptie = $_POST["foobar"];
$deleteOptie = $_POST["DeleteData"];
$table = "wp_cool_plugin_options";
$data_array = array('idOptions' => 1, 'isFoobar' => $foobarOptie, 'DeleteData' => $deleteOptie);
$where = array('idOptions' => 1);
$wpdb->update( $table, $data_array, $where );
And now it works like a charm 🙂