Generating graph for random walk given hitting time distribution
There are lots of different kinds of random walks (discrete vs continuous, weighted vs constant, classical vs quantum) so the exact approach will differ some what in each case, but I will attempt to outline a fairly general approach here.

Usually the evolution of that system will be given by $A^t$ for some matrix $A$, where $t$ is time. The exact nature of $A$ will depend on the type of random walk you have in mind, but in any case, we expect the system to be in a state $A^t v$ after time $t$, where $v$ denotes the starting state. The question then becomes: given a set of hitting times $\{t_i\}$, corresponding states $\{u_i\}$, and hitting overlaps $\{p_i\}$, can we find some matrix $A$ which satisfies $u_i^\dagger A^{t_i} v = p_i$ for all $i$? Certainly this will not always be the case (since we could have conflicts where orthogonal states are required at the same time), so I assume you are really interested in how to find $A$ when there does exist an $A$ satisfying these relations. 

EDIT: My previous answer was incorrect beyond this point, so I have replaced that portion.

I am not 100% sure of the most efficient way to find $A$. You have a system of equations in the enteries of $A$, and these can obviously be solved numerically using, for example, Buchberger's algorithm. However, this approach is not computationally efficient, and there may exist a more straight forward approach.

If you simply want find a transition matrix which approximates this, then you can use any number of optimization strategies, though these too may be inefficient.

Note, here I have assumed a very general model which should be true for all random walks, but it makes it hard to say anything concrete about how hard long it will take to numerically find a solution. If you have more information (i.e. is this a discrete time walk? Is it classical (I assume so)? Are weighted transitions probabilities allowed or do they have to be constant? Is this associates with some lattice (1d, 2d, or an arbitrary graph)?) Then you can possibly do much better. In the worst case, though, this problem seems to be NP-hard (since you get an integer programming problem if you restrict the transition probabilities to be multiples of some constant).