The 4350 isn't a bad choice, it uses about 15 watts of power which is about the same as your integrated graphics, you could move up to a 4650 though which uses about 35 watts of power and still be in a safe zone for better performance.
A dirt cheap 4350 will run after all my mom runs a 6600gt 512mb with a 250w and that is with a 65nm conroe that is overclocked from 1.6ghz to 2.4ghz with out dipping below 11.4v one the rail. All though the psu is well cooled and sturdy built for an oem.
By the way what model is that cpu?
Yes it should work. Get pcwizard first and load it up then go to the voltage and temperature monitoring section and look what is the voltage on the 12v rail. If it is greater than 11.5~12v than you should have no issue.