Copdenvy's shellinabox runner allows you to start a shell in the browser in which you can compile and run your code. This is fine for applications that are console based. Is it possible to run an X Windows application in the cloud and see the graphical window?
Setting the DISPLAY variable in the shell to that of your Linux workstation could work, though it relies on you being in control of the local network for things like NAT and firewall settings. The attraction of the cloud IDE is that you only need a browser and an https connection.
Can you ssh -X into the Docker container from an X capable source? Where would you find the URL? How would you manage the ssh keys?