Updating WP Core should not affect the theme, but if the previous person edited theme files directly, updating the theme itself would. Copying everything to a local or staging site, then running the updates one by one and checking a few key pages of the site, should highlight any major problems and is a best practice no matter what type of themes and plugins you’re running.
If the last person edited theme files directly, it would be best to download an unaltered copy of the same version of that theme, and use a file comparison tool to identify which files they changed. (Example: BeyondCompare, or use version control to commit the last version in production – then commit the latest version of the parent theme, and you can then see exactly which files have changed and how.)
From there, you can then create a child theme that contains only those modifications. Then, you can update the parent theme, double-check everything, and potentially just make a tweak or two at most on your new child theme to bring it up to par.