I have a few suggestions:
- Change desktop environment. Changing to a more lightweight environment WILL speed up your machine (a lot), since Unity is one of the most bloated ones. I recommend Xfce for your machine. LXDE is a bit TOO lightweight and it's not worth it. Xfce is highly customizable and beautiful. No big cons. To install it in Ubuntu, run the command
sudo apt-get install xubuntu-core
(don't use xubuntu-desktop since it will install some possibly unwanted programs).
My computer is a 9 year old fighter with an Intel Celeron CPU @ 2.8GHz, 1512 MB of RAM and a GeForce 6200 GPU, and Xfce runs incredibly smooth.
- Install proprietary drivers
If you have NVidia or AMD drivers, look for the "Additional drivers" program, which comes preinstalled in Ubuntu. Install all proprietary drivers, reboot and feel the performance difference. Intel hardware usually won't need any extra drivers, since they are included in the kernel.
- In Xfce, disable Window compositing in the "Compositor" settings. You won't have that much eye candy such as shadows in the back of windows or transparency but the system will run better.