Will this PSU run HD 4670 1GB?

You only need about 5 amps at 12 volts for a 4670. So, yes, it should.

OTOH, the Codegen is junk and you should replace it. I'd recommend one of these:
Corsair 400CX (30 amp 12 volt rail)
http://www.newegg.com/Product/Product.aspx?Item=N82E168...

Thank you for the answer.
Also, will my AMD Athlon X2 3800+ with 3GB DDR2 RAM bottleneck HD 4670 a lot?

It will depend on the resolution, but the most likely answer is "yes, at least some."
I agree with jsc about the PSU though; I would not trust Codegen. If the Corsair is not available to you, a 380W Antec Earthwatts would be another excellent choice.