There is a section in the admin guide that should help you with the steps to migrate RStudio Connect from one server to another, including the existing application content:
https://docs.rstudio.com/connect/admin/files-directories.html#server-migrations
The steps include installing the corresponding R versions and system packages on the new server.
If you plan to migrate Connect at regular intervals in the future, you might consider mounting the variable application data directory in Connect on an NFS server and using an external Postgres database to have a couple of less items to move during your next migration:
https://docs.rstudio.com/connect/admin/high-availability.html