First of all backup your current website’s version (files/folders/database)
Supposed you have ZenCart on yourdomain.com pointed to /home/USERNAME/public_html
So, we will upgrade your ZenCart to /home/USERNAME/public_html/shop to prevent any issue with your current installation.
Create sub domain shop.yourdomain.com and point it to /home/USERNAME/public_html/shop
Create if needed shop folder under /home/USERNAME/public_html/
mkdir -p shop
cd shop/
Download the latest version
wget https://sourceforge.net/projects/zencart/files/CURRENT%20-%20Zen%20Cart%201.5.x%20Series/zen-cart-v1.5.5f-12312017b.zip
unzip zen-cart-*
cd zen-cart-*
mv * ../
cd -
rm -rf zen-cart-v1.5.5f-12312017b/
Copy configure.php files from old installation to your new one
cp -rp /home/USERNAME/public_html/ADMINFOLDERNAME/includes/configure.php /home/USERNAME/public_html/shop/admin/includes/configure.php
cp -rp /home/USERNAME/public_html/includes/configure.php /home/USERNAME/public_html/shop/includes/configure.php
Create SQL dump of your current database. Create new database USERNAME_shop. Import data from old database to your new one
mysqldump USERNAME_oldwebsite > USERNAME_oldwebsite.sql
mysqladmin create USERNAME_shop
mysql USERNAME_shop < USERNAME_oldwebsite.sql
Update just copied configure.php files with new paths/domain/database
Copy images from old website
mv /home/USERNAME/public_html/shop/images /home/USERNAME/public_html/shop/images_old
cp -rp /home/USERNAME/public_html/images/ /home/USERNAME/public_html/shop/images
At this step ensure you are on the latest supported PHP version 7.x at this moment
Go to http://shop.yourdomain.com/zc_install and click “Upgrade“. If something doesn’t work double check you did all right in configure.php files
Remove zc_install/ folder
Rename admin/ folder as ZenCart will strongly suggest to do
Probably you will need to copy your templates
Reinstall the latest versions of your modules
Test everything before you decide to move it from shop.yourdomain.com to yourdomain.com