How to do an FFT of a line profile obtained by imagej?
It appears there is no straightforward way to do it from the UI. After looking through ImageJ docs and finding out how to start the interpreter, I've come up with something that is essentially this:

macro "FFT1D Tool - C099A8877099L0088" {
    profile = getProfile();
    fft=Array.fourier(profile,"None");
    s=newArray(lengthOf(fft));
    for (i=1;i&lt;lengthOf(fft);i++) {
        s[i]=2*PI/i;
    }
    s[0]=0
    Plot.create("Fourier spectrum of the line profile","wavenumber","FFT(line)",s,fft);
    Plot.show();
}


If placed in StartupMacros.fiji.ijm in the fiji folder, on the next start of fiji, the FFT1D tool will show up in a toolbar menu and, after a line was drawn over the image, a click on the toolbar button will execute the macro. 

A caveat: DFT works great for periodic functions and input line sizes that are powers of two. Manually drawing over the image is likely going to result in a non-power-of-two, aperiodic array. Running this through a DFT will result in Gibbs effect which will ruin the accuracy of the transform.    Window functions (substitute "None" for one of the available window functions) can help ameliorate this effect. It helps if the intensity of the image decays towards the endpoints of the profile line.

Also, check this example of using the Array.fourier function.