Usage of the word "itself"

"Itself" is not used in this way in British or American English. I suspect that it may be used in other Englishes: Irish and Indian come to mind, but I'm not sure.

It does seem to fill a hole, as there is not an obvious word in UK or US which fulfils the same role, of emphasising the time.

I would say "just yesterday" and "just this morning": these are not quite the same meaning, as they are emphasising how recent the time was, whereas I interpret the original remarks as emphasising the time for some reason but not necessarily its recency.

No, the two examples you cite don't sound too good.

And in any case "itself" doesn't carry the meaning of 'already'. It is used to place emphasis on the 'thing' to which it refers. So, reworking your first example:

"I got this information from the note itself, not from the explanation from Dave or John"

Pre-edit, just so Joe's comment makes sense:

"So was it Dave or John who told you about that message I left?"
"Neither, I read the note itself."

if the usage of "itself" is itself wrong, what would be the correct usage or alternative to express the meaning mentioned in the brackets

What you've already put in the brackets is pretty good, though you'd want some commas in there and a few tweaks to make it read better:

I already read the note, yesterday.

and

I have already met the doctor, this morning.