  Fast radio bursts (FRBs) are a new kind of extragalactic radio transients.
Some of them show repeating behaviors. Recent observations indicate that a few
repeating FRBs (e.g., FRB 121102) present time--frequency downward drifting
patterns and nearly 100$\%$ linear polarization. Following the model of
\citet{dai 2016} who proposed that repeating FRBs may originate from a
slowly-rotating, old-aged pulsar colliding with an asteroid belt around a
stellar-mass object, we focus on the prediction of time--frequency drifting and
polarization. In this scenario, the frequency drifting is mainly caused by the
geometric structure of a pulsar magnetosphere, and the drifting rate--frequency
index is found to be $25/17$. On the other hand, by considering the typical
differential mass distribution of incident asteroids, we find that an asteroid
with mass $m\gtrsim 10^{17}~{\rm g}$ colliding with the pulsar would contribute
abundant gravitational energy, which powers an FRB. A broad frequency band of
the FRBs would be expected, due to the mass difference of the incident
asteroids. In addition, we simulate the linear polarization distribution for
the repeating FRBs, and constrain the linear polarization with $\gtrsim$ 30$\%$
for the FRBs with flux of an order of magnitude lower than the maximum flux.
