Identifying whether or not a cyclostationary signal is noisy or not using the cyclic autocorrelation
You're on the right track. The cyclic autocorrelation can be estimated by inverse Fourier transformation of the cyclic periodogram. If $X_T(t, f)$ is the Fourier transform of your data block (indexed by time $t$), and $\alpha$ is your desired cycle frequency, you can form the cyclic periodogram by shifting the Fourier transform to the left and to the right by $\alpha/2$ (not by $\alpha$ as you state in your question):

$$I_x^\alpha(t, f) = \frac{1}{T} X_T(t, f+\alpha/2) X_T^*(t, f-\alpha/2)$$

Then just inverse Fourier transform that cyclic periodogram. You'll have to be careful about the case where the cycle frequency $\alpha$ is not equal to an integer number of FFT bins.