When precessing nuclei absorb electromagnetic radiations, am I supposed to consider the wave nature or particle nature of light?
The nuclear spin states in an atom are excited by the radio frequency radiation and one spin state moves to the other when a photon is absorbed. 

The precession part is not so much about individual spins but the magnetisation of the sample which is the vectorial addition of all the nuclear magnetic dipoles; precession is motion about the permanent magnetic field. (Think of the magnetisation rather like a vertical rotating pendulum if you want.) As circular polarised rf is used the spins are excited coherently and all add up 'in phase', hence a large magnetisation is produced that then undergoes precession. Loosing this phase in the magnetisation corresponds the the T2 decay time and loosing population excess above equilibrium to the T1 time. It happens that the precession frequency is the same as the transition frequency.

(An individual spin angular momentum is a complicated nutation type motion as it experiences a force from the rf field which is at right-angles to he permanent magnetic field.)