If you’re keen to use Kubernetes, I recommend asking in a Kubernetes specific support channel.
IMO, what you want to do is not that difficult to do manually. No package manager required. Download the WP .zip file to your local computer and extract the .zip file. Upload the extracted files to the desired server location via FTP. Then run /wp-admin/install.php. Fill in the form and WP will finish the setup for you.
If you want an URL like example.com/blog/ to lead to your home page, and other pages would have an URL such as example.com/blog/contact/, these will work without any further adjustments. However if you want just example.com/ to lead to the home page when WP is installed in /blog/, you’ll need to adjust your hosting account to take the naked domain name to a subdirectory instead of /html/.
Using the official WordPress image is fine, but for a custom path you’ll need to adjust WORDPRESS_CONFIG_EXTRA and set WP_SITEURL/WP_HOME to /blog. A reverse proxy rewrite can also handle it cleanly.
You can use the official WordPress + Apache image, but WordPress doesn’t natively support sub path installs easily. A better approach is to deploy it at root and handle /blog with reverse proxy or Ingress rewrite in k8s. Ketch can help simplify routing and deployment management for this setup.
