I know I'm a noob here on the forums, but you won't have an issue putting a larger PSU in that machine.
As an engineer, I know that your machine only draws as much power as it needs. If your computer only needs 350W, then that's the peak value the new 650W PSU will provide.
Now if you throw in a new, thirsty GPU, then you need to consider ventilation in the case due to the GPU itself and the PSU dissipation.