It sounds like what you want is for your Docker container to respond to a URL like http://localhost/some/random/path
by somehow specifying that path in the Docker --port
option.
The short answer to that is no, that is not possible. The reason is that a port is not related to a path in any way - an HTTP server listens on a port, and serves resources that are found at a path. Note that there are many different types of servers and all of them listen on some port, but many (most?) of them have no concept of a path at all. For example, consider an SMTP (mail transfer) server - it often listens on port 25, but what does a path mean to it? All it does is transfer mail from one server to another.
There are two ways to accomplish what you're trying to do:
write your application to respond to particular paths. For example, if you're using the Express framework in your node application, create a route for the path you want.
use a proxy server to accept requests on one path and relay them to a server that's listening to another path.
Note that this has nothing to do with Docker - you'd be faced with the same two options if you were running your application on any server.