In this paper, we study a one-dimensional interacting anyon model with a Stark potential in the finite size. Using the fractional Jordan Wigner transformation, the anyons in the one-dimensional system are mapped onto bosons, which are described by the following Hamiltonian: \begin{eqnarray} \hat{H}^{\text{boson}}=-J\sum_{j=1}^{L-1}\left(\hat{b}_{j}^{†}\hat{b}_{j+1}e^{i\theta \hat{n}_{j}}+h.c.\right)+\frac{U}{2}\sum_{j=1}^{L}\hat{n}_{j}\left(\hat{n}_{j}-1\right)+\sum_{j=1}^{L}{h}_{j}\hat{n}_{j}, \end{eqnarray} where $\theta$ is the statistical angle, and the on-site potential is $h_{j}=-\gamma\left(j-1\right) +\alpha\left(\frac{j-1}{L-1}\right)^{2}$ with $\gamma$ representing the strength of the Stark linear potential and $\alpha$ being the strength of the nonlinear part. Using the exact diagonalization method, we numerically analyze the spectral statistics, half-chain entanglement entropy and particle imbalance to investigate the onset of many-body localization (MBL) in this interacting anyon system, induced by the increasing of the linear potential strength. As the Stark linear potential strength increases, the spectral statistics transition from a Gaussian ensemble to a Poisson ensemble. In the ergodic phase, except for $\theta=0$ and $\pi$, where the mean value of the gap-ratio parameter $\left\langle r\right\rangle\approx 0.53$, due to the broken time reversal symmetry, the Hamiltonian matrix becomes a complex hermit one and $\left\langle r\right\rangle\approx 0.6$. In the MBL phase, $\left\langle r\right\rangle\approx 0.39$, which is independent of $\theta$. However, in the intermediate $\gamma$ regime, the value of $\left\langle r\right\rangle$ strongly depends on the choice of $\theta$. The average of the half-chain entanglement entropy transitions from a volume law to an area law, which allows us to construct a $\theta$-dependent MBL phase diagram. The time evolution of the half-chain entanglement entropy $S(t)$ increases linearly with time in the ergodic phase. In the MBL phase, $S(t)$ grows logarithmically with time, reaching a stable value that depends on the anyon statistical angle. The localization of particles in a quench dynamics can provide evidence for the breakdown of ergodicity and is experimentally observable. We observe that with the increasing of $\gamma$, the even-odd particle imbalance changes from zero to non-zero values in the long-time limit. In the MBL phase, the long-time mean value of the imbalance is dependent on the anyon statistical angle $\theta$. From the Hamiltonian $\hat{H}^{\text{boson}}$, it can be inferred that the statistical behavior of anyon system equally changes the hopping interactions in boson system, which is a many-body effect. By changing the statistical angle $\theta$, the many-body interactions in the system are correspondingly altered. And the change of the many-body interaction strength affects the occurrence of the MBL transition, which is also the reason for MBL transition changes with the anyon statistical angle $\theta$. Our results provide new insights into the study of MBL in anyon systems and whether such phenomena persist in the thermodynamic limit needs further discussion in the future.