No, replacing the CPU won't help the situation, as it is most likely fine. Mobile cpu's are made to take a lot of heat, so unless a fan on or near the heat sink failed, it's fine. You can order a new motherboard for it though and transplant the processor and such over to the new one. In the meantime, if you know of any PC repair business nearby, you can take it there and they can most likely diagnose the specific problem and tell you what you need to order.
Either that or get in touch with Sony and see what they would want to fix it, especially if it isn't under warranty.http://www.sparepartswarehouse.com/