We investigate the sensitivity of the Nash equilibrium of constrained network aggregative games to changes in exogenous parameters affecting the cost function of the players. This setting is motivated by two applications. The first is the analysis of interventions by a social planner with a networked objective function while the second is network routing games with atomic players and information constraints. By exploiting a primal reformulation of a sensitivity analysis result for variational inequalities, we provide a characterization of the sensitivity of the Nash equilibrium that depends on primal variables only. To derive this result we assume strong monotonicity of the mapping associated with the game. As the second main result, we derive sufficient conditions that guarantee this strong monotonicity property in network aggregative games. These two characterizations allows us to systematically study changes in the Nash equilibrium due to perturbations or parameter variations in the two applications mentioned above.