Besides looking at your inbox for any messages that WP might send you (they don’t always get sent), look at the error.log file in the site root and wp-admin folders. That will tell you where the error is.
If it is a plugin, then rename that plugin’s folder temporarily, then try the update again. You could also rename the entire plugin folder to temporarily disable all plugins. And you might consider reverting to one of the ‘twenty’ themes to do your update.
Once you have updated WP (and updated your PHP version, which may also be the problem; contact your hosting place for instructions on that), you can move back to your old theme. Then make a new plugins folder, and move each plugin folder individually to the new plugins folder. After each move, check the admin page, and do the ‘update’ check. Repeat for each plugin. (And update your theme, if needed.)
Your issue shows the importance of keeping WP – core, themes, and plugins – updated. Less problems that way. (And the importance of updating your PHP version.)