"Today" in the past

Let's say I'm talking about some day in the past. In formal writing, I would use:

Earlier that day, I had lunch with my boss.

But is the following also correct?

Earlier today, I had lunch with my boss.

Clarification: I'm not talking about something that happened today (as of the date when I'm writing), but some day in the past... last week, 3 months ago, etc.

Can I still write this?

It was the 1st of April, 2006. Today had been the worst day of my life...

Solution 1:

Earlier today would be understood as referring to something happened this day (today has origin from a Old English word that means on this day); earlier that day is referring to a day that you mentioned before.

To reply to your other question,

It was the 1st of April, 2006. Today had been the worst day of my life.

seems awkward, as today is not understood to mean on this day (the original meaning).

In narrative, an event that is happened in the past is narrated as it is the present, as in:

It is the 1st of April, 2006. Today will be the worst day of my life.

Outside that specific context, I would write

It was the 1st of April, 2006. That day had been the worst day of my life.

Solution 2:

“Earlier today” is a totally correct way to refer to a point in time between the beginning of the day and the current time. Because it refers to a moment in the past, it can be used with the past tense, as you did in your example.