Ubuntu server is a nice platform for server-related activities. Here is a short tutorial of how I updated my most current version to the latest available by rstudio.org. Here is how I got it going.
If this is your first install, you need to grab the gdebi stuff
sudo apt-get install gdebi-core
Next download the latest deb from rstudio. I typically like to try out the preview release, often stable enough to get what you want done while at the same time highlighting the latest features. When writing, it was the 1.2.1321 version.
wget https://s3.amazonaws.com/rstudio-ide-build/server/trusty/amd64/rstudio-server-1.2.1321-amd64.deb
Make sure to check the md5sum!
If you already have it running, stop it with
sudo rstudio-server stop
then install the new version
sudo gdebi rstudio-server-1.2.1321-amd64.deb
This went out and grabbed some other libraries and installed everything for me then turned it back on. Since I had it already installed, that was the end of it. If this is the first time you are installing it, you can configure it following the installation guide here.