Structural and magnetic properties of a quasi-one-dimensional spin-$1/2$ compound NaVOPO$_4$ are explored by x-ray diffraction, magnetic susceptibility, high-field magnetization, specific heat, electron spin resonance, and $^{31}$P nuclear magnetic resonance measurements, as well as complementary textit{ab initio} calculations. Whereas magnetic susceptibility of NaVOPO$_4$ may be compatible with the gapless uniform spin chain model, detailed examination of the crystal structure reveals a weak alternation of the exchange couplings with the alternation ratio $alphasimeq 0.98$ and the ensuing zero-field spin gap $Delta_{0}/k_{rm B} simeq 2.4$~K directly probed by field-dependent magnetization measurements. No long-range order is observed down to 50,mK in zero field. However, applied fields above the critical field $H_{c1}simeq 1.6$,T give rise to a magnetic ordering transition with the phase boundary $T_{rm N} propto {(H - H_{rm c1})^{frac{1}{phi}}}$, where $phi simeq 1.8$ is close to the value expected for Bose-Einstein condensation of triplons. With its weak alternation of the exchange couplings and small spin gap, NaVOPO$_4$ lies close to the quantum critical point.
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