Kirigami, the ancient eastern art of folding and cutting paper, has inspired novel techniques for the fabrication of 3D microstructures which self-fold spontaneously from flat templates, without the need of manual intervention. These folded structures offer a plethora of new physical functionalities compared to flat materials. Understanding how and how long it takes for the planar templates to fold into the desired 3D shape is then of growing interest and importance. Here we use a regular pyramid with N lateral faces as the target structure, which folds due to thermal fluctuations from a N-pointed star template. We assume that this template is pinned through the base at a substrate, preventing misfolding. For simple geometries, the rotational motion of the lateral faces is consistent with Brownian movement. Folding occurs through a sequence of binding events between lateral faces at a specific closing angle. We propose a lattice model to study the dynamics of self-folding, mapping the angular position of each lateral face into a unidimensional diffusing particle. The time evolution of the system is studied using a kinetic Monte Carlo approach. From independent numerical results we show that the average folding time varies non-monotonically with the number of lateral faces N, leading to the existence of an optimal value of N for fast folding. We also find that, if the template starts from a planar configuration, this time average is governed by a simple scaling law with the closing angle when N is large enough. Applying concepts from first passage processes and order theory, these numerical findings are supplemented with analytical results. The characteristic time scale of the folding process is also affected by the binding mechanism. We compare the average folding time of several initial configurations of the net and find that stacking the lateral faces on top of the base yields the fastest results. The slowest times are associated with either a planar conformation or when lateral faces are configured in opposite sides of the angular space. Lastly, we additionally study a system which folds in suspension, which allows for the possibility of defects in the target structure when lateral faces bind on both sides. Under these conditions, numerical independent results show that the optimal number of lateral faces for fast folding increases when compared to the case on a substrate. The folding efficiency, defined as the fraction of samples without defects, decreases with N and a specific balance between folding time and efficiency needs to be considered depending on the priorities.