I met him __________ Monday.

The correct preposition to complete the sentence "I met him __________ Monday" is "on". Prepositions of time are essential for indicating when an event occurs, and their usage depends on the specificity of the time reference.

• "On" is used for specific days of the week (e.g., on Monday, on Tuesday) and specific dates (e.g., on December 25th, on my birthday). Since "Monday" is a specific day, "on" is the appropriate choice.
• "At" is typically used for precise times (e.g., at 3 o'clock, at noon, at midnight) or for holiday periods (e.g., at Christmas). It is not used for days of the week.
• "In" is used for longer periods of time such as months (e.g., in July), seasons (e.g., in summer), years (e.g., in 2023), or parts of the day (e.g., in the morning, in the afternoon, in the evening, but not "at night"). It is too broad for a specific day.
• "For" indicates duration (e.g., for two hours, for a week) and does not fit the context of specifying when an event happened.

Therefore, the grammatically correct sentence is: "I met him on Monday."