a good cache method on the cpanel server?

Although the cPanel server limits some caching options, there are a number of options, but based on the main problem of the "first to byte" response, I would probably consider these options first.

If you want to speed up the on-site cPanel server response, the LiteSpeed ​​web server is an excellent upgrade with caching capabilities. An Nginx upgrade may be the case if the cost of the Litespeed technology is prohibitive.

PHP opcache is another thing that is unmistakable and the use of PHP 7.

Memcache and Redis is also something else worth looking at if site code can work with it.
We have also achieved some great results with Lack, but the setup is not so straightforward, but it can be done.

If some performance improvements can be achieved by using a MySQL server by updating to MariaDB, it all depends on your use case, not on the configuration optimization.

Make sure gzip compression is enabled, static content is cached by the browsers, and the css / js files are minimized to save transmission time.

As @Mhostingin A cloudflare or other CDN network for static content is likely to have an impact as well.

There is very little information about which site it is, but these are just a few options that you can examine within yourself. If the performance of a server is still dissatisfied, it may be useful to hire a sysadmin to check server performance and best practices. This is also a question of the hosting provider
may have some suggestions for your current use case.