(1) Am I approaching in right way?
It seems not.
I suppose you have ~ 90 sample points and just plot the absolute values of fft(y).
The left half of the "plot" corresponds to the negative frequency, i.e., it is just the phase that is different from the right hand side.
The two impulses correspond to the frequency of the sine wave (positive and negative).
If you have no idea about negative frequency, you should hit the book to learn the basics of FFT.
(2) Is that the impulse making the noises?
Threfore no, it is not.
What kind of noise did you add? If it were a Gaussian, the shape of it in the frequency domain is also Gaussian.
(3) How can I cut them off?
In Signal processing manner, you should apply filter(s) to it.
For simplicity, you can just remove the certain elements from the signal directly. This effectively applies a rectangle shaped filter, which causes some other side effects.
