In general, several (even not 'equivalent') matrices can have the same characteristic polynomial. Thus, it does not make sense to ask whether the process can be reversed, as there (seems to be) no unique algorithm to associate a matrix to a given characteristic polynomial.
However, given a characteristic polynomial (to be specific, $x^4-x^3-1$), is it possible to find an integer matrix such that both:
- all entries are non-negative,
and
- the sum of the integers in each column is the same?
no, it's not possible. If the sum of the integers in each column is the same (integer) $n,$ then the vector $(1,1,1,1)^T$ is an eigenvector (not of $A$ but of $A^T$) with eigenvalue $n.$
However, the polynomial $x^4 - x^3 - 1$ does not have any integer roots.