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Quantitatively understanding of the dynamics of an active Brownian particle (ABP) interacting with a viscoelastic polymer environment is a scientific challenge. It is intimately related to several interdisciplinary topics such as the microrheology of active colloids in a polymer matrix and the athermal dynamics of the in vivo chromosome or cytoskeletal networks. Based on Langevin dynamics simulation and analytic theory, here we explore such a viscoelastic active system in depth using a star polymer of functionality $f$ with the center cross-linker particle being ABP. We observe that the ABP cross-linker, despite its self-propelled movement, attains an active subdiffusion with the scaling $\langleΔ\mathbf{R}^2(t)\rangle\sim t^α$ with $α\leq 1/2$, through the viscoelastic feedback from the polymer. Counter-intuitively, the apparent anomaly exponent $α$ becomes smaller as the ABP is driven by a larger propulsion velocity, but is independent of the functionality $f$ or the boundary conditions of the polymer. We set forth an exact theory, and show that the motion of the active cross-linker is a gaussian non-Markovian process characterized by two distinct power-law displacement correlations. At a moderate P{é}clet number, it seemingly behaves as fractional Brownian motion with a Hurst exponent $H=α/2$, whereas, at a high P{é}clet number, the self-propelled noise in the polymer environment leads to a logarithmic growth of the mean squared displacement ($\sim \ln t$) and a velocity autocorrelation decaying as $-t^{-2}$. We demonstrate that the anomalous diffusion of the active cross-linker is precisely described by a fractional Langevin equation with two distinct random noises.