  We have performed $^{63}$Cu nuclear magnetic resonance/nuclear quadrupole
resonance measurements to investigate the magnetic and superconducting (SC)
properties on a "superconductivity dominant" ($S$-type) single crystal of
CeCu$_2$Si$_2$. Although the development of antiferromagnetic (AFM)
fluctuations down to 1~K indicated that the AFM criticality was close, Korringa
behavior was observed below 0.8~K, and no magnetic anomaly was observed above
$T_{\rm c} \sim$ 0.6 K. These behaviors were expected in $S$-type
CeCu$_2$Si$_2$. The temperature dependence of the nuclear spin-lattice
relaxation rate $1/T_1$ at zero field was almost identical to that in the
previous polycrystalline samples down to 130~mK, but the temperature dependence
deviated downward below 120~mK. In fact, $1/T_1$ in the SC state could be
fitted with the two-gap $s_{\pm}$-wave rather than the two-gap $s_{++}$-wave
model down to 90~mK. Under magnetic fields, the spin susceptibility in both
directions clearly decreased below $T_{\rm c}$, indicative of the formation of
spin singlet pairing. The residual part of the spin susceptibility was
understood by the field-induced residual density of states evaluated from
$1/T_1T$, which was ascribed to the effect of the vortex cores. No magnetic
anomaly was observed above the upper critical field $H_{c2}$, but the
development of AFM fluctuations was observed, indicating that superconductivity
was realized in strong AFM fluctuations.
