It is known that the CTAB/NaSal aqueous solution forms wormlike micelles when the concentration is higher than a certain value and shows the flow-induced structure change when both the shear rate and shear strain exceed respective critical values. In the present work, the start-up behavior of this surfactant solution in a two-dimensional abrupt contraction channel driven by a constant pressure was investigated. At a low driving pressure, the flow rate increases gradually and reaches an equilibrium state. When the driving pressure is higher than a certain value, the flow rate rises at a certain elapsed time. The elapsed time showing increase in flow rate shortens with increasing driving pressure. In the entry area of the narrow channel, a pair of wedge shape opaque regions appears. From the comparison with the observation in Couette flow of the same solution, it is found the opaque regions are caused by shear induced structure and the viscosity of the regions is higher than the other region. The regions grow up and make the other region with low viscosity narrower. The opaque regions suddenly break and disappear when the flow rate changes. Flow instability is induced by a sudden change of flow states. These phenomena are characteristic of the fluid showing the flow-induced structure change, whereas ordinary polymer solution do not exhibit such behavior.