If you have already backed up your data, you can make mistakes, that costs only a little time
A clean install would be the easiest way to install, particularly if you accept the default installation that the installer wants. It should give you one root partition with the operating system and space for your files, and one swap partition, to be used, when the RAM cannot contain all the data, that needs to be remembered.
Do you have a 32-bit alias i686 version or a 64-bit alias amd64 version of Lubuntu? Or have you not downloaded anything yet?
And what about the present system, Ubuntu 13.04, 32-bits or 64-bits?
An alternative, particularly if you have not downloaded lubuntu yet, is to install the desktop environment only:
sudo apt-get install lubuntu-desktop
This will install the desktop environment with or without the extra tools, that come with lubuntu, and you might not need, because you have tools from Ubuntu.
sudo apt-get install lubuntu-desktop --no-install-recommends
You can select session (lubuntu or ubuntu) at the login screen (click the circular icon in the attached picture).
The other desktop's files take some space on the hard disk drive, but they are not running, so you will see a big difference in performance between lubuntu and ubuntu.