Lets say that one wants to major in economics for undergrad school. But he wants to obtain a PhD in math in graduate school. If he self studies math on the side, can he still get into good graduate schools?
It would depend on what specifically you've studied in your undergrad days, and what field of physics you wish to enter into. I don't think it's that common for an engineering graduate to switch to physics, but is more common the other way around.
For the maths/economics question, again it depends upon how much maths you've studied, and what field you wish to enter into. I think your chances are greater if you wanted to go into applied maths or stats rather than pure, though.