Suppose there is a neutron star with a hole in the middle. Through this hole, scientists shoot a laser beam in such a way that it will pass through the gravity well of the neutron star but will not be bent by it.
I imagine that when photons are going in the hole, they blueshift, but after leaving they redshift (due to curved spacetime?). But how about the time they take from one side to the other? Will it be the same as if the neutron star wasn't there? If not, how do I calculate it?
The time taken for the light to traverse the neutron star will be longer than it would be if the neutron star wasn't there. This because the velocity of light as measured by an observer far from the neutron star is reduced by the gravitational field of the neutron star. This has been confirmed experimentally by measurements of the Shapiro delay. In this experiment radar waves are bounced off the surface of a massive object and they are found to take slightly longer to return than expected because of the decrease in the speed of light near the massive body.
If you're interested in finding out more about how the speed of light is affected by gravity have a look at Speed of light in a gravitational field? As described in that question outside the neutron star the speed of light is given by the equation:
$$ v = c \left(1-\frac{2GM}{c^2r}\right) \tag{1} $$
where $M$ is the mass of the neutron star.
Inside the neutron star the equation for the speed of light is much more complicated, and indeed in general there is no simple equation for it. If the neutron star had a constant density then the speed would be given by the Schwarzschild interior metric and we'd get:
$$ v = c \left(\frac{3}{2}\sqrt{1-\frac{2GM}{c^2R}} - \frac{1}{2}\sqrt{1-\frac{2GMr^2}{c^2R^3}}\right)\sqrt{\left(1-\frac{2GMr^2}{c^2R^3}\right)} \tag{2} $$
where $R$ is the radius of the neutron star. To calculate the time required take equation (2) for the speed and integrate it to get the traversal time. I'll leave that as an exercise for the reader.
