In a classical problem for the stopping of a diffusion process $(X_t)_{t \geq 0}$, where the goal is to maximise the expected discounted value of a function of the stopped process ${\mathbb E}^x[e^{-βτ}g(X_τ)]$, maximisation takes place over all stopping times $τ$. In a Poisson optimal stopping problem, stopping is restricted to event times of an independent Poisson process. In this article we consider whether the resulting value function $V_θ(x) = \sup_{τ\in {\mathcal T}({\mathbb T}^θ)}{\mathbb E}^x[e^{-βτ}g(X_τ)]$ (where the supremum is taken over stopping times taking values in the event times of an inhomogeneous Poisson process with rate $θ= (θ(X_t))_{t \geq 0}$) inherits monotonicity and convexity properties from $g$. It turns out that monotonicity (respectively convexity) of $V_θ$ in $x$ depends on the monotonicity (respectively convexity) of the quantity $\frac{θ(x) g(x)}{θ(x) + β}$ rather than $g$. Our main technique is stochastic coupling.