If your netbook can boot from USB you are all set to go. Any distro should work. Easiest way is probably to plug your external HD into a full desktoop or laptop computer then boot that computer with a LiveCD of the distro you want to install e.g Linux Mint.

Once that computer boots into the LiveCD desktop. Click the install icon and be sure to select your connected USB external drive as the drive to install the OS. Make sure to select the same drive as the place to install the bootloader.

Alternatively. INstall to a USB stick right from your netbook and then boot the netbook into the distro and install to the USB drive. That way you have a bootable USB stick. (you may not even need to install to the drive!) see this thread