Electret Mic preamp for sampling and FFT
Digital Signal Processors can do the FFT in real time. They have a particular architecture which means that certain instructions (MAC - multiply accumulate) can be done very quickly.

But then this turns in to an embedded software project too as well as the electronics design, if you're ok with that.

You can get development boards from companies such as Texas Instruments which contain a processor and break-out box arrangement for experimentation.