There is possible a firewall problem. First of all check your domain is configured right in your ‘live’ dashboard then, if not helps, contact your hosting provider.
To test the ports use this simple script in php:
$host = 'smtp.outlook.com';
$ports = array(25, 587);
foreach ($ports as $port)
{
$connection = @fsockopen($host, $port);
if (is_resource($connection))
{
echo '<h2>' . $host . ':' . $port . ' ' . '(' . getservbyport($port, 'tcp') . ') is open.</h2>' . "\n";
fclose($connection);
}
else
{
echo '<h2>' . $host . ':' . $port . ' is not responding.</h2>' . "\n";
}
}