The neutron emission in coincidence with the fragments have been investigated for the ⁴⁸Ti + ²⁰⁸Pb reaction populating the near superheavy compound nucleus ²⁵⁶Rf at an excitation energy of 57.4 MeV. The National Array of Neutron Detectors facility is used for a precise determination of the prescission (M^pre_n ) and postscission(M^post_n ) neutron multiplicities as a function of fission observables. A moving source fitting procedure has been adopted to deduce M^pre_n and M^post_n . The variation of M^pre_n with the mass-split and total kinetic energy (TKE) of the fission fragments have been studied to understand the fission dynamics of ²⁵⁶Rf. It is observed that M^pre_n increases from the value of 1.66 ± 0.07 to 2.23 ± 0.07 with transition from the asymmetric to the symmetric mass region. M^pre_n is also found to increase with the decrease in TKE, which is probably due to the neutron emission during the acceleration time of the fission fragments in this heavy system. The experimental results for neutron multiplicity have also been compared with the theoretical predictions from the statistical model calculations.From this comparison, the value of reduced dissipation strength for the ²⁵⁶Rf nucleus is found to be (13.0 ± 1.0)× 10²¹ s^-1 and a fission delay time of (67.3+5.3-3.9) × 10^-21 s has also been estimated. For the spontaneous fission of ²⁵⁶Rf, the extracted average neutron multiplicity M^sf_n is found to be 4.4 ± 1.0 which is in good agreement with the recently reported value for the ^258,260Rf isotopes.