I vividly remember that

[nininaz]

Is the use of "that" optional or mandatory?

I vividly remember that one day, when I was just about to leave the office, my phone rang, and as I picked it up I just heard someone shouting swear words at me.

 

[bhaisahab]

It's not strictly necessary, but I like it.

 

[yi-ing]

Is the use of "that" optional or mandatory?

I vividly remember that one day, when I was just about to leave the office, my phone rang, and as I picked it up I just heard someone shouting swear words at me.

Do the bold part fit into the definition below?

https://www.macmillandictionary.com/dictionary/british/be-about-to-do-something

 

[Matthew Wai]

The definition fits your sentence.

 

[jutfrank]

It's not completely clear to me what you want to say.

When you start talking about a memory like this, you could begin:

I vividly remember when I was...


Also, we often start stories like this:

One day, when I was...

Here, one day is just a way to show that you're going to tell a story. It doesn't matter which day.


Now, you could also possibly begin like this:

I vividly remember one day, when I was...

Here, one day means one particular day. In this case, you cannot use that, because one day is the object of the verb remember.

 

[GoesStation]

Does the bold part fit into the definition below?

See above. The bold part​ is singular so it requires the singular form of the verb.