Although important for detecting entanglement, the transpose operation cannot be directly realized in laboratory because it is a nonphysical operation. It is, however, possible to find an approximate transpose operation using the method known as the structural physical approximation (SPA); recently, SPA-based implementations of the transpose and partial transpose have been demonstrated for a single-qubit [Phys. Rev. A 83, 020301(R) (2011)] and an entangled two-qubit system [Phys. Rev. Lett. 107, 160401 (2011)]. In this work, we expand SPA-transpose to a three-dimensional quantum system: a qutrit. The photonic qutrit state is encoded in the polarization, and path degrees of freedom of a single-photon and the SPA-transpose operation, which is based on measurement and preparation of quantum states, is implemented with linear optics. Our work paves the way toward entanglement detection for higher-dimensional quantum systems.