I need to update a clients WordPress site to WP 4, however I'm a little nervous (it is a big, expensive site). How likely is it that something will go catastrophically wrong?
ALWAYS take a database backup and file backup of the website before you update. If something does go wrong you can revert.
If you want to avoid as much interruption as possible, set up a local or development copy of the site and perform the update there. When you are done, back up your development server and launch that on the live site again.
If 30 minutes of downtime won't kill you, it might be quicker to update directly. But if something does go wrong, your website will be offline until you restore the backup.
If you do not have many 3rd party plugins and your theme is recently updated, it should go pretty smoothly. The older your theme/plugins, and the more you have, the more likely you will experience issues.
WordPress does a pretty good job at providing backwards compatibility, but plugin/theme authors are a different story.
Making backup and testing on development enviroment is very important but but as important is to be familiar with the code of the site and try to read the release notes or even the trac tickets to have an idea what areas deserve extra testing. In addition check plugins and theme compatibility with the new version, try to read or even ask at the support forums.
This can turn out to be a lot of work, but the last thing you want to happen is to make an upgrade only to discover a week later that all the paypal transaction had failed but for some reason were marked as successful.
And yes, from the wording of your question it is obvious that your client do not expect to be billed for a lengthy testing session so as the others said, backup, backup, backup, and cross your fingers that if there will be major bugs it will be discovered short time after the upgrade when you can still fix them by restoring from the backup.