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Depending on what that private container is serving up, an API gateway might work better. Tyk is one that's written in golang. It has a rest API and hot reload, so it should be able to handle your use case of dynamically allocated ports.

http://tyk.io

You can use service discovery tools like Consul or Etcd. Basic idea that on container/app boot, you register your app to service discovery service (you give it IP and current port), and it stores information about your all running apps.

After you can use consul Nginx integration (it will dynamically generate Nginx config on all updates, and will restart it).

Tyk mentioned here also have Consul service integration and much more.

Sounds like Traefick.io mentioned above should do the trick there. Looks like an interesting tool...

Nginx can proxy to a server specified in a variable. All you have to do is define that variable though perl or lua to get port dynamically from somewhere, like a file.

eBay/fabio or Traefik.io both fit that description. Linkerd and Envoy probably do too but I have less experience with those.