It may be possible, but I don't know that I would risk it unless you are willing to upgrade the PSU if necessary. I wouldn't bother upgrading both for an old P4 system.
My recommendation would be to step down to an HD 6570 DDR3. It has the same power requirements as your current card so it will work for sure, and it will still be a nice upgrade over the X1300. Also, I see no reason to get a card that will push your power requirements. The P4 and PCIe 1.0 will hold you back anyway.