Finding minimum possible cost of road network between cities with distance from capital condition
The MST indeed adresses conditions 1 and 3 but not conditions 2. The solution of the global problem (as shown by your example) is not the MST but still a tree.
Let's call $T$ the solution for the input graph $G$. Let's also call $T_i$ the solution for the problem $G_i$ which is the subgraph of $G$ containing vertices with range index lower or equal to $i$ (I will use the term &quot;range index&quot; to prevent confusion with the vertex indexing used for Prim's algorithm).
Trivially $T_0$ and $G_0$ contain only the capital. Now what about $G_1$ ?
The cities having a range index of 1 cannot use largest index cities to connect captial. Thus $T_1$, which is by the way MST of $G_1$, is a subgraph of $T$.
$T_{i+1}$ is built the same way using $T_i$ and $G_{i+1}$. The edges of $T_i$ are kept and only the new edges to connect the cities having range index $i+1$ are added using any MST algorithm.