If you use a standalone microcontroller (ARM/Atmel/PIC etc), you typically (these days) use an in-circuit programmer. Then you can either use a debugger or "crash and burn" methods (probably with a terminal interface) to develop your code.
I can get that with the correct HDL code, you might integrate a microprocessor or controller with your FPGA hardware design.
But how do you then get your code onto it? and how do you debug it? do you also synthesise a way to load compiled code? or is the code somehow build into the bitstream that you load onto the FPGA? is there a standard method, or does it vary by platform?