Let ${\xi(t); t\geq 0}$ be a normalized continuous mean square differentiable stationary normal process with covariance function r(t). Further, let $$ \rho(t)=\frac{(1-r(t))^2}{1-r(t)^2+r'(t)|r'(t)|} $$ and set $$ \delta=\frac{1}{2}\wedge \inf_{t\geq 0} \rho(t). $$ We give bounds which are roughly of the order $T^{-\delta}$ for the rate of convergence of the distribution of the maximum and of the number of upcrossings of a high level by $\xi(t)$ in the interval [0,T]. The results assume that r(t) and r'(t) decay polynomially at infinity and that r''(t) is suitably bounded. For the number of upcrossings it is in addition assumed that r(t) is non-negative.