Ubuntu is a recent concept to me... I've only heard of it when vmware came out with their free player and ubuntu as the browser appliance! (so I've never actually installed it)
But what will run and not run...
Bassically if it said i386, yep, you can boot that on a 386 computer... If it ever runs Gnome or KDE, that is an entirely different story! Post the specs and people who use the different distros's will tell you if it might work or not!
For now, I recommend getting a small live cd just to see that the computer will function correctly and if it does, then go for a huge dvd install... Slax or DSL are both good small live cd's
I have the KillBill Edition
there are multiple version of this, including a live cd version!