Adding cname names for folders

I need to achieve following via cost effective was as possible , without revealing my IP, thanks.

I am giving my services to my users via client.example.com and there are pages like

client.mysite.com/blog
client.mysite.com/blog/content/ 
client.mysite.com/docs/ 

etc.

I want to allow users to allow their domains to point to this subdomain.

so they can choose between any of the 1 option below :

  client.com -> client.mysass.com
  sub.client.com -> client.mysass.com
  client.com/sub/ -> client.mysass.com

and pages should work automatically like

 client.com->  client.mysass.com
 sub.client.com/blog -> client.mysass.com/blog
 client.com/sub/blog -> client.mysass.com/sub/blog

Also, I use Elastic beanstalk in amazon to deploy my react application with nginx (docker image ). Before I start I want to know if this is possible.I also don’t want to give fixed ip address to my clients, just in case if I lose that IP.

I learned that we can do this by custom hostname for SAAS in Cloudflare , apex a

but can anyone explain how to do this client.com/blog -> client.mysass.com/blog

You could set up a Page Rule to perform a redirect.

https://support.cloudflare.com/hc/en-us/articles/218411427-Understanding-and-configuring-Cloudflare-Page-Rules-Page-Rules-Tutorial-

https://support.cloudflare.com/hc/en-us/articles/4729826525965-Configuring-URL-forwarding-or-redirects-with-Page-Rules

i can do that if that is my domain , how does my client does it ? when his apex is pointed to some other IP and some other dns provider ? and make his folder point to my default origin server (fallback.mysass.com)

Can some one explain in my client point of view.