Firstly, there are a few issues with a time-dependent potential, $V(x,t)$. Namely, if we apply Noether's theorem, the conservation of energy may not apply. Specifically, if under a translation,
$$t\to t +t'$$
the Lagrangian $\mathcal{L}=T-V(x,t)$ changes by no more than a total derivative, then conservation of energy will apply, but this resricts the possible $V(x,t)$, depending on the system.
We often treat each Schrödinger equation case by case, as a certain system may lend itself to a different approach, e.g. the harmonic oscillator is easily solved by employing the formalism of creation and annihilation operators. If we consider a time-dependent potential, the equation is generally given by,
$$i\hbar\frac{\partial \psi}{\partial t}=-\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial \mathbf{x}^2} + V(\mathbf{x},t)\psi$$
Depending on $V$, the Laplace or Fourier transform may be employed. Another approach, as mentioned by Jonas, is perturbation theory, whereby we approximate the system as a simpler system, and compute higher order approximations to the fully perturbed system.
Example
As an example, consider the case $V(x,t)=\delta(t)$, in which case the Schrödinger equation becomes,
$$i\hbar \frac{\partial \psi}{\partial t} = -\frac{\hbar^2}{2m}\frac{\partial^2 \psi}{\partial x^2} + \delta(t)\psi$$
We can take the Fourier transform with respect to $t$, rather than $x$, to enter angular frequency space:
$$-\hbar\omega \, \Psi(\omega,x)=-\frac{\hbar^2}{2m}\Psi''(\omega,x) + \psi(0,x)$$
which, if the initial conditions are known, is a potentially simple second order differential equation, which one can then apply the inverse Fourier transform to the solution.
