Here is Deusovi's answer in a simpler language. All credit goes to him.
Alice is guaranteed to win. She can win by ensuring that by the end of the 7th turn, the sum is not divisible by 3.
She can do this by :
Case 1: If sum is divisible by 3 by the end of the 6th turn. Alice can put a number 3x+1 or 3x+2. At least one of them has to be available.
Case 2: If sum is not divisible by 3 by the end of the 6th turn. Alice can put 3x if it is available. If 3x is not available then that means that 3x was the last move. Then she can put either 3x+1 or 3x+2.
And here is why ensuring that the sum is not divisible by the end of the 7th turn, guarantees Alice a win :
Look at the following figure. It shows the 2nd last turn of the game(t=2) i.e Alice's last turn. l=0 means that Bob put a 3x number on his previous turn, etc. s=0 means that the sum is 0 mod 3 after Bob's turn i.e the sum is divisible by 3.
Thus, Bob can win in t=3(his second last turn, overall 8th turn) if
He can make the sum modulo 3=0 by playing a number divisible by 3.
He can make the sum modulo 3=1 by playing a number 3x+1.
He can make the sum modulo 3=2 by playing a number 3x+2
All the above are only possible if the sum is divisible by 3 by the end of the 7th turn. So, Alice just needs to ensure that the sum is not divisible by 3 by the end of the 6th turn.