As you mentioned that you have done some customization to the theme. I am assuming the customization has done inside child theme instead of the main theme.
If you are not using the child theme, then please move the custom code that you have in the main theme to the child theme and take a full backup of your website first.
Then now you can update your theme and WordPress.
Just got the Dashboard -> Updates you will find the WordPress update button and the list of items which needs the update. From there you can update the Theme and WordPress.
I am really sorry but unfortunately we are not allowed to update or fiddle with core functioning of the customers website. Here is something that you can do to update the website without hampering live website:
Take a complete site backup and copy the website to a staging server.
After that, you need to install child theme and move the changes you have made directly into the theme file to child theme. This is to preserve the changes you have done, reason being the moment you click update button WordPress will start replacing old files with the new ones.
Next step would be to update Theme and plugins. You can take a look at the step by step instructions mentioned in our update guide. Link for the same has been shared by my colleague in previous thread.
If you face any issues, please let us know and we will help you out with the same.
If everything goes well, you can either perform the same steps on the live website or move the website that’s there on the staging server to production server.