1. I will be busy that day.

2. I will be busy on that day.
(Which one is right?)

3. I met the man that Saturday.
4. I met the man on that Saturday.
(Which one is right?)

All are correct, but 1 and 3 are more common.

Both versions of each sentence are correct, but the usage depends on the context and personal preference.

1. "I will be busy that day."
2. "I will be busy on that day."

In sentence 1, "that day" is treated as a specific day that has already been mentioned or is understood by both the speaker and the listener. In this case, "that day" acts as a noun phrase and can stand alone without the preposition "on."

In sentence 2, "on that day" includes the preposition "on," which is often used to specify when an event or activity will take place. Both versions are grammatically correct, but the inclusion of "on" might provide more clarity for some readers or listeners.

3. "I met the man that Saturday."
4. "I met the man on that Saturday."

Similarly, both versions of sentences 3 and 4 are grammatically correct, but the usage depends on personal preference and the context of the conversation.

In sentence 3, "that Saturday" acts as a noun phrase and refers to a specific Saturday, which has already been mentioned or is understood. The preposition "on" is not necessary in this case.

In sentence 4, "on that Saturday" includes the preposition "on," indicating the specific day when the meeting took place. This version with the preposition may provide more clarity and specify the time frame more explicitly.

To determine which version to use, consider the context and style of your writing or speaking. If you want to be more specific and precise, including the preposition "on" can help clarify the time element in your sentence.