I have a script that will query remote database at times. But the connection speed from my local server to the remote one is too slow. I'd like to know whether there is some kind of script which can download data from a remote database and update them to local one regularly. If so, the query response will be much faster.

I've read about master-slave setup. But there isn't an option to edit my.cnf in cPanel. So is it possible to use a script to automatically back up database to local server and update them in local database?

Try MySQL replication. This allows one database to be replicated to one or more database servers. Updates to the remote database can be synchronous or asynchronous. Asynchronous is particularly useful if the network connection or remote server goes down. The transactions are queued up and played to the database when the connection or server resumes.