How to update MySQL data within intranet from MySQL data in cloud?
MySQL has a built-in functionality called MySQL replication that allows the synchronisation between 2 servers, by applying on a "slave" the operations that have been done on the "master". If no writes are done on the slave, that works pretty well and, while it has its issues, it is widely used by most MySQL users (Facebook, Twitter, etc.) for high availability, load balancing and/or testing/maintenance operations.

It is very easy to setup and has little performance impact, as it is asynchronous by default. That makes possible to perform the replication in real time or in batches, as desired.

Please note that while all "Infraestructure as a service" allows replication, some "database as a service" providers do not allow to do it outside of their servers (for example, RDS only allowed it in MySQL 5.6 servers).

If you need to write to the slave (your intranet server), that would be more complex, as you have to make a plan on how to merge the results and modify the replication settings or use an alternative technology.