The success of e-commerce business offering same-day delivery depends on customer satisfaction. To speed up deliveries and lower costs, some companies have been using private individuals as non-dedicated drivers to perform pickup and delivery tasks for online customers. Such delivery systems are known as crowd-shipping. Customers have come to expect an accurate estimate for the delivery times of their online orders. The coordination of online deliveries with private individuals is done by a crowd-shipping platform. In this paper, we focus on the estimation of pickup and delivery times. This is a challenging job because not only are the requests unknown and submitted dynamically, but so is the pool of drivers, i.e. delivery capacity. We model the problem as a Markov decision process and integrate it into a simulation study. To improve the estimates that can be done by a naive policy, we propose two policies that use lookahead: one with a fixed lookahead horizon and one with a dynamic. Our numerical experiments demonstrate that a lookahead policy with dynamically adjusted horizon outperforms the other two policies in terms of estimation accuracy, which is up to 19% higher in some instances.