Sampling and reconstruction of bandlimited graph signals have well-appreciated merits for dimensionality reduction, affordable storage, and online processing of streaming network data. However, these parsimonious signals are oftentimes encountered with high-dimensional linear inverse problems. Hence, interest shifts from reconstructing the signal itself towards instead approximating the input to a prescribed linear operator efficiently. In this context, we propose a novel sampling scheme that leverages the bandlimitedness of the output as well as the transformation whose input we wish to approximate. We formulate problems to jointly optimize sample selection and a sketch of the target inverse mapping, so when the latter is affordably applied to the sampled output signal, the result is close to the desired input. The developed sampling plus reduced-complexity processing pipeline is particularly useful for streaming data, where the linear transform has to be applied fast and repeatedly to successive response signals.