Alice can win using the following strategy: if possible, place a domino in the leftmost available place in the middle row. If it is not possible to play in the middle row, place a domino directly above or below one previously placed in the middle row.
First, I claim that Alice will be able to play at lease 17 dominos in the middle row. To see why, note that there are initially 99 places a domino can go in the middle row. Call such a place blocked if one or both of its squares is occupied by another domino. Each domino Alice or Bob places blocks at most 2 previously unblocked places (so that Bob can block at most 4 places on each of his turns). 16 turns by Alice and 16 turns by Bob lead to at most $16\cdot 2+16\cdot 4 =96$ of the 99 places being blocked, so Alice can still find a place to play in the middle row on her 17th turn.
Each domino placed in the middle row makes space for two more dominos, one above it and one below it, and Bob cannot interfere with these spaces. So after playing 17 dominos in the middle row, Alice is guaranteed to be able to find a play for the next 34 turns. This gives Alice a total of at least 51 plays.
Finally, Bob will not be able to play 51 times. In 51 turns, Bob would need to place 102 dominos. Each domino Bob places must be in a different column, and there are only 100 columns on the board.
In fact, Bob cannot play in any of the (at least) 34 columns containing one of Alice's middle row dominos, so Bob will actually be able to take a total of at most 33 turns.