The OSRM docker quick start provides a great explanation on how to set up the container: https://hub.docker.com/r/osrm/osrm-backend/
Yet - for Docker on Windows minor changes were necessary for me (otherwise I'd get "File not found" or "Permission" errors).
This is how it worked for me:
Examples of getting certificates from Let's Encrypt working on Apache, NGINX and Node.js servers.
I chose to use the manual method, you have to make a file available to verify you own the domain. Follow the commands from running
git clone https://github.com/letsencrypt/letsencrypt
cd letsencrypt