Cut this apple . . . . . . that knife.

Cut this apple with that knife.

The preposition "with" is used to indicate the instrument or tool used to perform an action. In this case, the knife is the tool used to cut the apple, so "with" is the correct choice.

Option A: by is incorrect because it refers to the agent performing the action, not the tool used.

Option C: in is incorrect because it suggests location or direction rather than the tool used to perform the action.

Option D: into is incorrect because it indicates the result or direction of an action (e.g., cut into pieces), but here we are focusing on the tool used to cut the apple.

Conclusion: The correct answer is Option B: with, as it properly indicates the use of a knife as the tool to cut the apple.