The simplest way to say it is 'that day'.
What did you do on June 1st?
I went to work [on] that day.
You usage of 'on the day' is not exactly correct. If you want to indicate you did something on a specific day, you would specify the day.
On July 6th I went to the store.
On the day of July 6th I went to the store.
I can think of no usage for "in the day". It simply isn't said. If you want to indicate you did something while the sun was in the sky you would say, "in the daytime" or "during the day".
We went to the zoo in the daytime and at night we got dinner.
During the day we went to the zoo.
If you want to combine the date and daytime constructs you would say.
On August 3rd we went to the zoo during the day and out to eat in the evening.