The general answer to your final question,
How do you solve the Schrödinger equation for a generic time-dependent Hamiltonian?
is somewhere between "it depends", "it's complicated", "good luck", "with brute-force numerics", and "you don't".
But if the question is: are there commonly-used settings in which the solutions of the time-independent Schrödinger equation are useful for understanding the behaviour driven by a time-dependent Hamiltonian? then the answer is yes.
It happens very frequently that we have systems with Hamiltonians of the form
$$
\hat H(t) = \hat H_0 + \hat V_I(t),
$$
where $\hat H_0$ is (i) very complicated but (ii) time-independent, and $\hat V_I(t)$ is time-dependent but much simpler, and typically represents some form of interaction. One good example of such a setting is that of a many-electron atom driven by a laser field, where $\hat H_0$ is a complex multi-electron beast which involves lots of kinetic energies, spin-orbit couplings, and electrostatic interactions, plus a lot of correlation involved in its eigenstates, and where $\hat V_I(t)$ is just the dipole moment operator of each electron taken separately from the others, and multiplied by a simple sinusoidal function of time.
In this case, there is going to be a lot of hard work involved in solving for the eigenstates of $\hat H_0$, but this only needs to be done once, typically in advance. And once you have that, then you can have a much easier time in treating the effect that $\hat V_I(t)$ has in promoting transitions between the different eigenstates of $\hat H_0$.
If you want to see how this works in a general setting, then you should look up Time-Dependent Perturbation Theory in your favourite advanced-QM textbook.
Of course, it bears mentioning that this is not the only approach that fits the description you've given but that is too broad to provide an exhaustive list.
