hmmm...350W? It depends on the components. If you are planning to use only the iGPU, 350W should be enough.
350W is not enough to support dedicated GPU such as R9 270x, etc.
Usually I would ask:
How old is the PSU? From what brand? is it built inside your E5200 rig?
Since from LGA775 to 1155 will be a huge overhaul, I would simply replace the PSU with a new one something from Corsair, Seasonic, BeQuite!, PPC, etc. (any well-known brand), you will need about 500-550W