How do I install gcc 4.8.1 on Ubuntu 13.04?

Use the mirrors listed Here and download the 4.8.1. The process is pretty straightforward. I would recommend to use this Procedure to complete your installation.

As you may know GCC doesn't support "make uninstall" and it has been suggested that you install GCC into a directory of its own and simply remove that directory when you do not need that specific version of GCC any longer. Hope this helped. Cheers

Edited: The Option 2:

I assume that you already have a former version of gcc, the easiest way could be adding PPA to your repositories and Update and upgrade you can have the latest version with no worries:

sudo add-apt-repository ppa:ubuntu-toolchain-r/test
sudo apt-get update

this will add the new PPA to the other sources.

Then unistall the alternative:

sudo update-alternatives --remove-all gcc 
sudo update-alternatives --remove-all g++

then:

sudo apt-get install gcc-4.8
sudo apt-get install g++-4.8

and as the alternative packages install :

sudo update-alternatives --install /usr/bin/gcc gcc /usr/bin/gcc-4.8 20
sudo update-alternatives --install /usr/bin/g++ g++ /usr/bin/g++-4.8 20
sudo update-alternatives --config gcc
sudo update-alternatives --config g++

at the end:

sudo apt-get update
sudo apt-get upgrade -y
sudo apt-get dist-upgrade

Hope this changes the --version ;)

If you want to keep your old gcc, as I do, then do this instead:

sudo add-apt-repository ppa:ubuntu-toolchain-r/test
sudo apt-get update
sudo apt-get install gcc-4.8 g++-4.8

Then to compile with older gcc, mine was 4.7:

gcc main.c

To compile with gcc 4.8:

gcc-4.8 main.c

You might find typing the extra -4.8 annoying, in which case follow the other answers given by raven and Amir. I quite like this method, as it gives me the choice to use a fall-back version (4.7) if I encounter a bug!