Is a time domain spectrum obtainable from a frequency domain spectrum?
Can you do the reverse?


Yes, mathematically the Fourier transform can be reversed. If we define the spectrum $S(\omega)$ as the Fourier transform of a time-domain signal $f(t)$, 

$$S(\omega) = \mathcal{F}[f(t)] = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}f(t)\mathrm{e}^{-\mathrm{i}\omega t}\,\mathrm{d}t$$

then the inverse Fourier transform is simply given by

$$f(t) = \frac{1}{\sqrt{2\pi}}\int_{-\infty}^{\infty}S(\omega)\mathrm{e}^{\mathrm{i}\omega t}\,\mathrm{d}\omega$$

Wikipedia has a page on this which is rather technical, but any decent book on Fourier transforms should cover this, so a mathematics book targeted at physicists/chemists may be easier to digest.


  What are the advantages and disadvantages of the frequency and time domain spectra?


The time domain signal (in NMR, "free induction decay") is for the most part, not particularly useful. It is the Fourier transform which makes it valuable for chemists, because the frequencies that appear in the spectrum are related to transitions between different energy states:

$$\Delta E = \hbar\omega$$

In general, probing these energy levels is the main point of spectroscopy, regardless of what range of energies you are using.