Will I have any problems?

Hi there. Yes it will work, the PCI Express 2.0 is backwards compatible with the PCI x16 1.0 slot motherboards. However, PCI-e 2.0 cards on 1.0 motherboards may not be able to run at their max capacity due to 2.0 having twice the bandwith of 1.0 but a graphics card needs to be powerfull enough to use the extra bandwith, so with a HD4670 you shouldn't be able to notice a difference.

I believe your computer is ATX, if it is ATX then the power supply will fit your computer.
 
It'd probably be a good idea to measure your case just to make sure the new power supply will fit. Many pre-built PCs use proprietary hardware and the new one might not fit. Better to be safe than sorry.