Upgrading from WP 3.6 to 4.3.1

I didn’t see a category for WP updates so am posting here. Feel free to move it if need be.

I have a client who has a problem with his posts not going to Facebook. But his Jetpack, or any other plugin, will not update, and I’m assuming it’s because the version is so old. (He has not maintained things.) Will we incur problems with doing a straight update from WP 3.6 to 4.3.1 from the site??

A backup program is not installed and my attempt to install one also did not work as the php is 5.2 and BackWPup (the program I’m familiar with) needs 5.2.7. and also needs a later version of WP. I attempted to load php5.3 from the server, but this broke his WP site.

Any suggestions…. or should I go with the WP update?

The server has backups of the database, so we’re safe on that end. Are there files or folder that I should also download from the server for safekeeping before I attempt the update (in lieu of a backup program)?

Thanks