I have a walk through for deploying a Shiny app on Google Cloud Platform that you could perhaps adapt for AWS, at least the Dockerfile - guide is here. If you don't mind using GCP then there are a lot of helper functions to help too, such as gce_shiny_addapp()
Those Shiny apps are launched via a startup script (cloud config here) that is below:
docker run -v /home/gcer/library/:/library/ \
--name=shinyserver \
-p 80:3838 \
-v /home/gcer/shinyapps/:/srv/shiny-server/gcer/ \
-v /home/gcer/shinylog/:/var/log/ \
your-docker-container
Note putting the port on 80 (web) that may be useful, and where to put the logs.
A working Dockerised Shiny app demo is in the repo here
The logs are either in the volume you link to the logs within the Docker, or a good debug is to log into the Docker container via docker exec -it dockername bash which puts you in a shell within it, if its running. The logs you can browse to /var/log/shiny-server
Otherwise, if the docker image crashed on startup you can see via journalctl -u docker.service to get an idea of what caused the crash, although it sounds like your Shiny is running.
Its probably either the ports not being opened (in which case mapping to port 80 will fix it) or you haven't the expected dependencies in your Dockerfile. For that, I recommend containerit which you can point at your Shiny script and it will work out the dependencies for you and write your Dockerfile for you. Its magic.
Yours sincerely,
Mark