The Live USB cannot be updated or upgraded. What you have is an image of a DVD inside the USB. You will have to install Ubuntu into another drive to get all the benefits of a full install. The second drive can be another USB drive.See How do I install Ubuntu to a USB key? (without using Startup Disk Creator) for how to do a full install of Ubuntu in an USB drive. In particular, follow the answer by Cumulus007. Read the other answers too. The more you know the better it is.
If I understand you correctly, the computer currently does not have a hard drive. In that case, you need not worry about installing Ubuntu on the hard drive and wiping all the contents in the process.
You will need two USB drives. The Live USB can be 1GB. The second one should be at least 8GB or bigger as a complete install will take up more space. You will also need space for updating and upgrading the system, installing new software, as well as for your documents, pictures, music, etc. I have tried doing a full install in a 4GB USB and it was too small for the install to complete. I have a 16GB USB with full install.
There are two things you don't want in the full USB install.
First, Normally, every time Ubuntu boots, GRUB the bootloader checks all the hard drives in the computer to see if there any other operating systems (OS) and adds them to the boot menu, so that you can choose the OS you want to boot from. You want your USB to be portable. That is, you may want to take it to a different computer and boot your own Ubuntu there. Every time you do that GRUB will add the OS of that computer to the list. So, if you boot your Ubuntu USB using 4 computers with XP, Vista, 7, and 8. Later you may see the choice to boot from all four even though your own computer does not have a hard drive and none of these OSs.
To prevent this from happening, do the following after you boot from your new full install USB. Open a terminal by pressing Ctrl+Alt+T and type:
sudo chmod -x /etc/grub.d/30_os-prober
Second, Do not create a Swap partition during install. By default, Ubuntu installation may create a Swap partition. This will require advanced installation option called "Something Else" and specify what kind of partitions to create during installation. See How to partition a disk for installing Ubuntu? for some instructions on how to create the partitions. The answer is a bit old and the images may not match yours but you will get the idea.
You should also disable swapping after installation. See reducing disk writes: swapiness=0 or swapoff better?
Also see the Swap FAQ at Ubuntu Community Documentation.
Hope this helps