Once you have your data and settings backed-up, you should do a fresh install of Ubuntu 12.04.1 LTS on your PC. There may be initial issues with drivers, but this should mostly be resolved by doing an update.
This could be from a DVD or USB, whichever is easiest for you.
Once you have 12.04 LTS working, then try out (but do not immediately install) Ubuntu 12.10. This should also work on your PC, but will in turn be replaced by 13.04 (in April 2013), etc.
I have successfully run 12.04.1 LTS on an HP Mini netbook (Atom processor, 1GB RAM) or ex-corporate desktops (P4 2.8GHz, 1GB RAM).
You are unlikely to be able to use all of the 'new' GUI elements of 12.04 (or 12.10) with a laptop of that age, but it should work fine.