How to do sliding window multiplication in frequency domain
Is there a way to get my vectors Y1, Y2, Y3, etc... by computations in the frequency domain instead?

Yes. Rough outline: you take a full length FFT if the zero padded window. For each frame you can implement the time shift a multiplication with $e^jn\omega$ and then do a circular convolution and downsample. If you choose the window size well, you can do a down-sampled convolution instead.

And how do I know if it's faster/slower than just taking the inverse FFT of the frequency response and doing the steps described above?

It will almost always be slower. Convolution has computational complexity of $N^2$ whereas multiplication has only $N$. Saving a few short FFTs will not compensate of this.