First and foremost, you will need to create a bootable USB-drive with a Live installation of Ubuntu (or some other Linux).
HOWTO: Create a bootable USB-drive in Ubuntu
Just download the image you want from ubuntu.com, launch 'Startup Disk Creator' (already installed by default) and choose to install the image you just downloaded onto a drive of your choice. This drive can be a CD, a DVD, a USB or an SD card. Be aware that any data stored on the drive you choose will be erased.
You will have to provide your root password during the installation, in order to give the installer the necessary permissions required to write a boot-loader onto the drive.
You can also choose to create a persistence file, which will allow you to save settings between sessions.
HOWTO: Create a bootable USB-drive in Windows
There are a number of tools available in Windows to create bootable media, but my personal favorite is UNetbootin. It's pretty straightforward, and it can even download the appropriate Live CD image for you.
Once you have successfully created a bootable USB-drive, plug it into the computer you wish to install Ubuntu on and (re-)start it. On the Asus EeePC line of computers, you can enter the boot-menu by pressing the Esc-key when you start the computer. Which exact key is used for this differs between brands.
That should be all you need to do to create and boot into a Live USB session on a computer without an optical drive.