Call a dot a source if it has an outgoing arrow, a sink if it has an incoming arrow, and unused if it has no arrows. Every arrow goes from a source to a sink.

Suppose $n\ge 2$. At the end, no dot is unused: an unused dot could receive an arrow from any source. Also, every source must be connected to every sink, since any missing arrow between them would be legal.

Therefore, if there are $a$ sources and $b$ sinks at the end, then $a+b=n$ and exactly $ab$ moves have been played.

When $n$ is odd, one of $a,b$ is even, so the total number of moves is even. Bob makes the last move and wins, regardless of how either player plays. For $n=1$, Alice has no legal move, so Bob wins as well.