How much time does it takes an electron to tunnel through a barrier? I know that in quantum mechanics there is no "time operator", so such a question is ill-posed. Anyway if the tunneling is instantaneous, this would imply an information transmission faster than $c$. On the other hand, how could someone define such a "time"?
 A: Assume a particle moves in a potential, in one dimension, of the form
$V(x) = \infty$ if $x^2 > a^2$ else $V(x) = \gamma \delta(x)$,where $\gamma >0$. Let $E_0$ be the energy of the particle. Then $\langle v \rangle = \sqrt{\frac{2E_0}{m}}$
Frequency of collisions = $\frac{\langle v \rangle}{a}$
Frequency of tunelling = $\frac{\langle v \rangle}{a} * T$
$T$ is the transmission probability which can be calculated by solving the Schrodinger equation.
Time required for a particle to tunnel = $\frac{a}{\langle v \rangle * T} $
Edit: Corrected Typo. $V(x)=\infty$
A: I think what you're asking here is:

"If a particle is about to tunnel through a barrier, how long will it take to get from one side to the other?"

If so then you should consider how the particle is tunnelling through the barrier. It tunnels through the barrier because it's wavefunction "leaks" through the barrier - which means that it has a non-zero probability to be located outside of the barrier (i.e. it's already outside the barrier "when it starts to tunnel through it"). Because of this, the question unfortunately doesn't have a well defined answer. 
If you're asking:

"How long will it take until a particle is transmitted through a barrier in a 1-dimensional system with two confining potentials" 

See ramanujan-dirac's answer.
