10.04 LTS hasn't been supported in quite some time so I doubt that there's a direct upgrade path. You're probably going to simply have to back up your documents and install a newer version of Ubuntu from scratch. If you like the old Gnome interface, as opposed to the current Unity interface that Ubuntu uses, you might want to try Ubuntu MATE, which uses sticks to the older desktop environment.
Other alternative lightweight modern versions of Linux include Lubuntu, Xubuntu, ChaletOS, Bohdi Linux, Linux Lite and WattOS. I'd probably try each one of them by booting them from a USB key and then installing the one you like best.
Edit: each of those variants has a 32-bit version, which is the version you need, as the Dell Mini 9's processor (Intel Atom N270) is 32-bit only.