We propose and demonstrate a novel and compact optical fiber surface plasmon resonance (SPR) sensor for the simultaneous measurement of the refractive index (RI) and temperature. The sensor is fabricated by employing the eccentric-core fiber (ECF), which is polished to a V-shaped micro-structure optical fiber (MOF). One side of the MOF is used as the RI sensing channel, and the other side is used as the temperature sensing channel. Two sensing channels are independent without crosstalk. The sensor has high RI sensitivity up to the maximum of 3376 nm/RIU from 1.333 to 1.385 RIU, and high-temperature sensitivity up to -2.65 nm/°C from 20°C to 60°C. In addition, owing to the few-mode light in the ECF, the full width at half-maximum is smaller than other SPR sensors based on multimode fiber. This sensor will be extremely helpful to improve the RI measurement uncertainty caused by temperature disturbance.