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The conditional value of a stationary random process, given the level-upcrossing of another dependent stationary random process, is considered. Assuming that both processes are weakly non-Gaussian, an analytical approximation for the related conditional distribution is derived. It is based on a trivariate Edgeworth expansion truncated to non-Gaussian terms of lowest order, to which Rice's formula is then applied. As an application, the effect of level-upcrossing conditioning in second-order ocean waves is investigated. Upcrossing events are monitored for the sea surface elevation. The conditional distributions of different kinematic variables, given upcrossing, are considered for different sea-state configurations. Predictions from the analytical model are compared with numerical data obtained from Monte Carlo experiments. It is found that the analytical approximation provides conditional mean and variance in good agreement with numerical data, although moderate discrepancies appear for the sea states with the most severe wave cambers. Regarding the conditional skewness, given upcrossing, results are mixed, with significant discrepancies between the analytical approximation and numerical estimates, in a number of cases. An Edgeworth-type approximation is also provided for the upcrossing frequency and compared with Monte Carlo estimates; this analytical estimate is found to be accurate over a wide range of crossing levels.