Well, of course, you have a complete and full backup to start – DB and files. Has to be said if this is going to be a complete answer.
It is hard to tell quickly what the problem could be when a WP upgrade produces a failure like you describe. Most likely it is the theme or plugins causing the content to no longer display after the upgrade.
Personally, I would do this on a copy of the site first to avoid taking down the production site. If everything goes badly, the customer still has a functioning website.
Take your DB and file system copy and spin up a dev site either locally or isolated in the current hosting account (/dev/works just fine). Create a new DB of course. Do NOT simply link to the live site DB. Also turn on debug in wp-config.php like this:
define('WP_DEBUG', true);
Upgrade WP core, themes and plugins in your dev site copy. This may solve the “none of my posts show” as the new plugin will be running on WP 4+. With any luck you’ll be OK at this point. If your theme was written well the site should survive this upgrade with very minimal issues, related to plugins.
If there is a problem, the usual debug steps apply, switch themes, deactivate all plugins/reactivate them one at a time. These steps are necessary to find the specific block of code that is causing the failure by a simple process of elimination.
When you find the offending item, be it theme or plugin related, your next steps to solve that will be determined.