I am just wondering is there any propagation period after A-record for a domain was changed or changes will start taking effects instantly?

Sometimes it is immediately, sometimes there is a delay.

You can check up here: http://www.lookupserver.com/mx.php

Enter the domain name in the DNS Record Lookup, the A record will be under Type: A

you need to change the serial number from the dns zone once you make changes. this will allow the zone file to update its records.