Recently, the numerical methods for long-time dynamics of PDEs with weak nonlinearity have received more and more attention. For the nonlinear Schrödinger equation (NLS) with wave operator (NLSW) and weak nonlinearity controlled by a small value $\varepsilon∈ (0, 1],$ an exponential wave integrator Fourier pseudo-spectral (EWIFP) discretization has been developed (Guo et al., 2021) and proved to be uniformly accurate about $\varepsilon$ up to the time at $\mathcal{O}(1/\varepsilon^2).$ However, the EWIFP method is not time symmetric and can not preserve the discrete energy. As we know, the time symmetry and energy-preservation are the important structural features of the true solution and we hope that this structure can be inherited along the numerical solution. In this work, we propose a time symmetric and energy-preserving exponential wave integrator Fourier pseudo-spectral (SEPEWIFP) method for the NLSW with periodic boundary conditions. Through rigorous error analysis, we establish uniform error bounds of the numerical solution at $\mathcal{O}(h^{m_0} + \varepsilon^{2−\beta_{\tau^2}})$ up to the time at $\mathcal{O}(1/\varepsilon^\beta)$ for $\beta ∈ [0, 2],$ where $h$ and $\tau$ are the mesh size and time step, respectively, and $m_0$ depends on the regularity conditions. The tools for error analysis mainly include cut-off technique and the standard energy method. We also extend the results on error bounds, energy-preservation and time symmetry to the oscillatory NLSW with wavelength at $\mathcal{O}(\varepsilon^2)$ in time which is equivalent to the NLSW with weak nonlinearity. Numerical experiments confirm that the theoretical results in this paper are correct. Our method is novel because that to the best of our knowledge there has not been any energy-preserving exponential wave integrator method for the NLSW.