Reducing closed TSP to open with fixed starting vertex
Add two dummy vertices, $d_1,d_2$.  Add edges of weight 0 from all other vertices to $d_1$.  Add an edge of weight 0 from $d_1$ to $d_2$.  Add an edge of weight 0 from $d_2$ to your fixed starting vertex.  (If you need a complete graph, each missing edge can be replaced by an edge with weight $+\infty$ or some very large positive weight.)  Then any Hamiltonian cycle in the resulting graph will correspond to a Hamiltonian path in the graph that starts at your fixed vertex.