TSP when cost depends only on location in sequence
Your problem is the same as minimum-weight perfect matching in a bipartite graph. If there are $n$ cities, consider the bipartite graph in which one side consists of vertices $p_1,\ldots,p_n$, the other side consists of vertices $c_1,\ldots,c_n$, and the weight of the edge $(p_i,x_j)$ is the cost of city $j$ at position $i$.

A minimum-weight perfect matching in a bipartite graph can be found efficiently (this is known as the assignment problem). In particular, the problem is not NP-hard unless P=NP.