Recent advances in single-molecule chemistry have led to designs for artificial multipedal walkers that follow tracks of chemicals. We investigate the motion of a class of walkers, called molecular spiders, which consist of a rigid chemically inert body and several flexible enzymatic legs. The legs can reversibly bind to chemical substrates on a surface and through their enzymatic action convert them to products. The legs can also reversibly bind to products, but at a different rate. Antal and Krapivsky have proposed a model for molecular spider motion over regular one-dimensional lattices [T. Antal and P. L. Krapivsky, Phys. Rev. E 76, 021121 (2007).]. In the model the legs hop from site to site under constraints imposed by connection to a common body. The first time a leg visits a site, the site is an uncleaved substrate, and the leg hops from this site only once it has cleaved it into a product. This cleavage happens at a rate r<1, slower than dissociation from a product site, r=1. The effect of cleavage is to slow down the hopping rate for legs that visit a site for the first time. Along with the constraints imposed on the legs, this leads to an effective bias in the direction of unvisited sites that decreases the average time needed to visit n sites. The overall motion, however, remains diffusive in the long time limit. We have reformulated the Antal-Krapivsky model as a continuous-time Markov process and simulated many traces of this process using kinetic Monte Carlo techniques. Our simulations show a previously unpredicted transient behavior wherein spiders with small r values move superdiffusively over significant distances and times. We explain this transient period of superdiffusive behavior by describing the spider process as switching between two metastates: a diffusive state D wherein the spider moves in an unbiased manner over previously visited sites, and a boundary state B wherein the spider is on the boundary between regions of visited and unvisited sites and experiences a bias in the direction of unvisited sites. We show that while the spider remains in the B state it moves ballistically in the direction of unvisited sites, and while the spider is in the D state it moves diffusively. The relative amount of time the spider spends in the two states determines how superdiffusively the spider moves. We show that the B state is Markovian, but the D state is non-Markovian because the duration of a D period depends on how many sites have been visited previously. As time passes the spider spends progressively more time in the D state (moving diffusively) and less time in the B state (moving ballistically). This explains both the transient superdiffusive motion and the eventual decay to diffusive motion as t→∞.