You can find the model number by opening the laptop, and looking somewhere on the LCD bezel. It will tell you the model number (VGN-?????).
Yes, you can buy a universal notebook adapter for all Sony laptops, but it *MUST* be labeled as a "universal" AC/auto/air adapter. An iGo would be an example. However, they are more expensive than standard AC adapters from Sony.
And if your notebook will not power on, call Sony support and have them run you through troubleshooting.