To determine which player has a winning strategy in the given game, we need to analyze the possible moves and outcomes strategically. The setup on the calculator presents us with a logical and combinatorial game theory problem. The key to solving this is to determine which player can force a win by controlling the game flow.

### Strategy Analysis

1. Game Setup Description:

- The calculator is arranged in a 3x3 grid (ignoring the digit '0'), and moves are restricted to same-row or same-column digits, except the last digit pressed.
- The target is to avoid being the first player to reach or exceed a sum of 31.

2. Initial Strategy:

- Player $A$ starts by selecting any digit and then presses "+". The choice of this initial digit influences the subsequent moves available to both players.
- Player $B$, having knowledge of $A$'s move, selects a digit in the same row or column, ensuring it is different from the last digit chosen by $A$.

3. Player B's Winning Strategy:

- Response Moves for Player B:
- Player $B$ aims to maintain symmetry and control over the game's pace. By always keeping the sum defensive and avoiding reaching 31, $B$ can enforce decisions on $A$.
- For each number $X$ chosen by $A$, player $B$ should choose $10-X$, thus keeping increments under controlled pace.

4. Ensuring the Opponent Exceeds 31:

- The sum of choices by $A$ and responding moves by $B$ should always land just under critical thresholds (e.g., 10, 20, 30) before passing the calculator back to $A$.
- Player $B$ uses mirroring tactics to settle $A$ into a sum of 31, forcing $A$ into a position of inevitability.

### Conclusion

Through a strategic understanding of move responses and taking advantage of symmetrical gameplay, Player $B$ can maintain control over the game. Given that Player $B$ does not begin the game and can choose responses tactically to restrict Player $A$'s winning paths, Player $B$ possesses a definitive strategy to win.

Thus, the answer is:
$$\boxed{\text{Player } B \text{ has a winning strategy.}}$$