When the 256 MB on the graphics card run out the cards starts using the system RAM, so in a way this means your graphics card has more memory to use. However using the system RAM, instead of the VRAM integrated on the graphics board takes a lot of time. It will severly slow down you graphics card. Any 256MB card will already be super slow and will be slowed down further if the board runs out of its integrated memory.
I have 256 MB on my laptop GPU, and it will barely run even the low spec free to play games. It will improve your system a lot if you buy a cheap graphics card.
If a game specifies you need 1gb of graphics memory it means you need 1gb of on board graphics VRAM. So buying 2GB of RAM will not help with this problem. Still if you only have 1GB of actual RAM, then buying more will improve your systems performance. Buying a cheap graphics card will also greatly increase the performance.