64 bit only

in: /etc/apt/sources.list
add: deb http://cran.rstudio.com/bin/linux/ubuntu precise/

sudo apt-key adv --keyserver keyserver.ubuntu.com --recv-keys E084DAB9

sudo apt-get update
sudo apt-get upgrade

sudo apt-get install r-base
sudo su - \
-c "R -e "install.packages('shiny', repos='http://cran.rstudio.com/')""
sudo apt-get install gdebi-core
wget http://download3.rstudio.org/ubuntu-12.04/x86_64/shiny-server-1.2.3.368-amd64.deb
sudo gdebi shiny-server-1.2.3.368-amd64.deb

Next Steps
Once installed, view the Administrator’s Guide to learn how to manage and configure Shiny Server. Stay up to date on RStudio Server news and software updates by subscribing above.

ref:

1. http://askubuntu.com/questions/218708/installing-latest-version-of-r-base

2. http://www.rstudio.com/products/shiny/download-server/