the only thing that matters is the sum of digits modulo 3 - if that sum is zero, Bob wins, and if not, Alice wins
So, let's work backwards:
Represent a state by three things: the sum so far ($s$), the last move played ($l$), and the number of turns remaining ($t$). In these charts, I will color each possible state red for it being a winning state for Alice, or blue for it being a winning state for Bob.
At $t=1$, it is Bob's turn; Alice wins if she had successfully prevented Bob from making the winning move.
At $t=2$, it is Alice's turn; she wants to make the move that would end on one of the "A" spots. If the sum is 0, she wants to play 0; if the sum is 1, she wants to play 1; and if the sum is 2, she wants to play 2. So she wins if she can make that move, and loses if she cannot.
At $t=3$, it is Bob's turn. If the sum is 0, he wants to play 0; if the sum is 1 or 2, though... he cannot land on any of the B spaces! Say the sum is 1. Playing 0 lands on the upper middle, playing 1 lands on the middle right, and playing 2 lands on the bottom left. Similarly, if the sum on Bob's turn is 2, any play will land the game state on an A space at $t=2$.
So, Alice wins with the following strategy: On the second-to-last turn, make the sum either 1 or 2 -- they can't both be blocked! Then, Bob will make a move, setting the new sum to $s$. On your final turn, the move $s$ will be open to you, and this sets the sum to $2s$. Now it's the final turn of the game, and Bob has to play $s$ again to win, but you just blocked it.
For the record:
Here is the entire transition graph for this game. Circles with dashed circumferences are legal starting states for Alice; yellow circles are the states that Bob wants to end the game with.