  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 $\alpha\simeq 0.98$ 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.
