Does it sound natural to say "Don't tell him yet. Let today pass first." ?
Does it sound natural to say "Don't tell him yet. Let today pass first." (Then you can tell him after today)

Question

Answer

Just leave it (=forget about it) for today, tell him tomorrow.
:)

Answer

I would recommend you say something more along the lines of:

Don't tell him yet. Give him a day.
Don't tell him yet. Let him know tomorrow.
Don't tell him yet. We'll see how today goes first.

Hope that helps :)