In a very recent experimental work (Gao et al 2018 Adv. Mater. 30 1707055), a graphene-like CuSe monolayer (ML) was realized. Motivated by this success, we performed first-principles calculations to investigate its electronic transport and photoelectronic properties. We find that the CuSe ML shows a strong electrical anisotropy, and its current-voltage (I-V) curves along the zigzag and armchair directions are noticeably different. The CuSe ML also displays a useful negative differential resistance (NDR) effect along the both directions when the bias is beyond 1.0 V. Moreover, it has a large photon absorption to orange light. Our study suggests that CuSe ML is a multifunctional material and has various potential applications in electrical-anisotropy-based, NDR-based, and even optical nanodevices.