I have set up my own server with R and shiny-server, and had already put some apps on that.
However, all apps have a port number like the :3838
in
You need to change a setting in the Shiny Server config file (change the port it listens on). See below (assuming you are running Ubuntu):
Find the shiny server config file (available in /etc/shiny-server/shiny-server.conf). Find the line:
# Instruct this server to listen on port 3838
listen 3838;
Change that to 80 (which is the default http port).
Then restart the shiny server service:
service shiny-server restart
All your apps will now be running through the :80 (or regular http port), which you don't need to mention when typing out the link.
NOTE: I'm not sure how this might affect other websites you might be hosting (I only run shiny/rstudio server on my box, so that wasn't something I needed to worry about).
For more information, see the shiny server documentation (Section 2.1 for your specific question)