  We establish the uniform error bound of an exponential wave integrator
Fourier pseudospectral (EWI-FP) method for the long-time dynamics of the
nonlinear Schr\"odinger equation with wave operator (NLSW), in which the
strength of the nonlinearity is characterized by $\varepsilon^{2p}$ with
$\varepsilon \in (0, 1]$ a dimensionless parameter and $p \in \mathbb{N}^+$.
When $0 < \varepsilon \ll 1$, the long-time dynamics of the problem is
equivalent to that of the NLSW with $O(1)$-nonlinearity and
$O(\varepsilon)$-initial data. The NLSW is numerically solved by the EWI-FP
method which combines an exponential wave integrator for temporal
discretization with the Fourier pseudospectral method in space. We rigorously
establish the uniform $H^1$-error bound of the EWI-FP method at
$O(h^{m-1}+\varepsilon^{2p-\beta}\tau^2)$ up to the time at
$O(1/\varepsilon^{\beta})$ with $0 \leq \beta \leq 2p$, the mesh size $h$, time
step $\tau$ and $m \geq 2$ an integer depending on the regularity of the exact
solution. Finally, numerical results are provided to confirm our error
estimates of the EWI-FP method and show that the convergence rate is sharp.
