Phase angle and amplitude -
Since \$\vec{E}\$  only got a component in y-direction and \$\vec{B}\$  got only a direction in z-component that the \$\vec{k}\$ is either in -x or x-direction.


True. Imagine a simple wave

$$A(t)=\sin\left( \vec{k}\vec{x}-\omega t \right)$$

with \$\vec{k}=\begin{pmatrix}\pi/2\\0\end{pmatrix}\$ at \$t=0\$.

Draw a grid with x, y in the range [-4;+4] and calculate the field at each position. (25 position, but you'll see, you don't have to calculate 25 values). 

You will notice wave fronts parallel to the y-axis, because the scalar product \$\vec{k}\vec{x}\$ only depends on the x-coordinate. This means, the wave travels parallel to the x-axis, so parallel to \$\vec{k}\$.

What happens a little bit later, say for \$\omega t=\pi/2\$ ? The wave will be shifted to the right, i.e. it travels to the right. And what if you invert \$\vec{k}\$ ?

You'll see, the direction of \$\vec{k}\$ is the direction where the wave travels.


  About the second part: In general E and B are in phase, right? 


No, not in general but for a single free wave


  Since that's not the case here, can I assume that it results from a reflection on a surface? But how would I get to B0 and the phase change without explicit values for ω and such? 


No, it's not a reflection, because a reflected wave travels into the opposite direction. While the incident wave is \$\sin(kx-\omega t)\$, the reflected is \$\sin(-kx-\omega t)\$  (see above). Because the arguments of your sine and cosine function are equal, there is no reflection.

Also, if you think there's a reflection in B, why isn't there an evidence for a reflection in E?

The only thing one can see is that B is written with a (not yet defined) phase shift relative to E. This can either be expressed as sum of sine and cosine function with different factors, or by a sine function with that phase shift build-in. By the trigonometric identity 

$$\sin \theta + \sin \varphi = 2 \sin\left( \frac{\theta + \varphi}{2} \right) \cos\left( \frac{\theta - \varphi}{2} \right)$$

you can change between both representations. You also do not need to know \$\omega\$ for this.

I'm not yet sure what else you should do, may be because you didn't post the full question. In general, giving the E-field, you can calculate the B-field via Maxwell's equations, which will lead to the dependency between \$E_0\$ and \$B_0\$ was well as \$\phi=0\$ . It will also give you the helicity (if the wave moves away from you and the maximum of E points upwards, does the maximum of B point to the right or left?).



By the way: There is no phase shift for a single wave. It is interesting, that for a standing wave (sum of wave and reflected wave of same amplitude), E and B are phase shifted by 90°.