In the last few years, the fractional Schrdinger equation (FSE) received a lot of attention in various areas of physics. In particular, since Longhi suggested a scheme to explore the FSE in optics, the propagation of light beams in the framework of the FSE has been investigated extensively. Also, the beams described by special functions, such as Airy beam, Gaussian beam and Bessel beam, were paid much attention due to their novel properties and potential applications in the related physical fields. However, among the investigations, the propagation dynamics of Airy beam in the framework of the FSE with a combined potential was rarely involved.In the paper, based on the FSE with a combined potential consisted of periodic potential and linear potential, the optical Bloch oscillation and Zener tunneling of Airy beam are studied by numerical simulation. It is found that under the condition of weak transverse force, the initial incident Airy beam first reaches the edge of Brillouin region due to the refractive index gradient, and then be reflected to the center of Brillouin region due to the Bragg reflection, at which the total internal reflectionof the beam occurs. This eventually leads to periodic oscillation of the Airy beam alternating between Bragg reflection and total internal reflection, i.e., optical Bloch oscillation. Different from the propagation in free space and uniform space, the Airy beam with Bloch oscillation can well maintain the Airy shape even after propagating for a long distance. It is related to the energy band structure of the system, i.e., with the decrease of the Lévy index, the energy band width and oscillation amplitude decrease. In this case, the Lévy index can not only change the energy band structure of the fractional system, but also effectively control the optical Bloch oscillation. Furthermore, increasing the linear potential transverse force would lead the appearance of optical Zener tunneling at the edge of Brillouin region, but the decreasing of the Lévy index could suppress this phenomenon effectively.