If a flow is unsteady in Eulerian description is also unsteady in Lagrangian description? The flow is steady/unsteady in both of the approaches or how does that relation go?
 A: No. Remember that a lagrangian description tracks the motion of a specific particle on time, and the eulerian description tracks the motion that happens in a specific point in space. Think about about a bucket with a hole in the bottom that is being fed with water on top and leaking water on bottom. At some point the water level in the bucket will stop changing (Steady state) but all the particles of water will still be moving from the entrance of the bucket to the exit of the bucket.
The specific spatial region of the bucket will be unchanging(Steady in an eulerian sense) with time, but the water particles are still crossing the boundaries of the bucket(Unsteady in an lagrangian sense)
You can see this easily mathematically by the definition of material derivative:
$$\frac{df(x,t)}{dt}_{X=constant} = \frac{\partial{f(x,t)}}{\partial{t}}+ \frac{\partial{x_{i}(X,t)}}{\partial{t}}  \frac{\partial{f(x,t)}}{\partial{x_i}} $$
X(capital x) represents a specific particle; x(lower case x) represents a specific point in space and f is a function.
You can se that you can have $\frac{\partial{f(x,t)}}{\partial{t}}=0$, that is, steady eulerian flow, with $\frac{\partial{x_{i}(X,t)}}{\partial{t}} >0$, that is, "lagrangian unsteady flow" (The correct term is simply that the individual particles of fluid are moving)
In general you can only have steady lagrangian motion if $\frac{\partial{x_{i}(X,t)}}{\partial{t}} =0$ which means that no particles are moving and the system is static as a whole. EDIT: notice that the sense of "static" I employed here is with respect to motion, not overall change in time.
You can still change some property of a specific region without motion happening. For instance, you take a fluid and put it the bottle until its full and the start heating the bottle. The temperature of the fluid inside the bottle will change, but no motion is happening, so $\frac{\partial{f(x,t)}}{\partial{t}} >0$ but $\frac{\partial{x_{i}(X,t)}}{\partial{t}} =0$
