The su-exec can be used in alpine.
Do add it the package, if not already available, add the following to your Dockerfile
RUN apk add --no-cache su-exec
Inside your scripts you’d run inside docker you can use the following to become another user:
exec su-exec <my-user> <my command>
Alternatively, you could add the more familiair sudo package while building your docker-file
Add the following to your Dockerfile that’s FROM alpine
RUN set -ex && apk --no-cache add sudo
After that you can use sudo
sudo -u <my-user> <my command>