Not sure I understand all of this but
As long as you can send
some-new-app.com to your IP address and
some-old-app.com to the same address it should not be a problem
You could use a CNAME record to forward both of these to
myrouter.asuscomm.com which is your actual device
Once it gets there I assume you forward the traffic to your web host where it should still be receiving requests (Host: headers) for
some-old-app.com and
some-new-app.com.
Then your web server should be directing the traffic to the correct rails app
It’s a bit difficult to diagnose without more information but hopefully this is a pointer