$wpdb in php 5.5

There is no difference in this respect between PHP 5.3 and 5.5. The code doesn’t need to change.

However, you should still use prepare properly. Like so:

$query = $wpdb->prepare( 'UPDATE wp_building_plan SET sale_price = %s, available = %s, broker_name = %s, broker_email = %s,    broker_phone = %s WHERE id = %s', $sale_price, $available, $broker_name, $broker_email, $broker_phone, $unit_id );
$wpdb->query($query);

If you are seeing a difference between the servers, then the issue is probably with the database itself. Is the table on one of them read-only or some such thing? Does the table even exist? Check the new database using a tool like phpMyAdmin and see what’s up there.