Using Fourier series we can decompose any any signal into it's elementary signals but condition is that signal should be periodic and sinusoidal one.
Now, is it possible to decompose nonperiodic sinusoidal signal with Fourier analysis or any other method? If yes, how and if no, why?
No it is not possible. A Fourier series is periodical by definition, because a Fourier series is the sum of periodical functions having a same period.
There is another concept which is Fourier transform where the functions "to be transformed" are not supposed to be periodical.