Running RStudio Connect in a container is possible but, in general, is not recommended. Connect is designed to isolate and sandbox user applications. In essence, Connect runs its own "containers". To do so requires root, which means for Connect itself to run in a container requires privileged access.
To address your needs, I'd instead recommend recommend running Connect in a persistent server. Connect is designed to use and support multiple versions of R. This flexibility ensures that content can be "deployed and forgotten" by R users who invariably will use multiple versions of R. We recommend building R from source in a directory structure that contains the R version, for example:
/opt/R/3.4.1
/opt/R/3.3.3
/opt/R/3.2.5
Instructions: http://docs.rstudio.com/connect/admin/r.html
The admin guide also includes a list of suggested system dependencies for R packages:
Ubuntu - http://docs.rstudio.com/connect/admin/getting-started.html#installation-ubuntu
REHL / CentOS - http://docs.rstudio.com/connect/admin/getting-started.html#installation-redhat
If you wish to pursue the Docker container further, please contact your Customer Success rep.
Best,
Sean