Just say, in four days (or three days, or five days,) from now.

The day after the day after the day after ... tomorrow It's usually easier to say a day, anything after 'the day after tomorrow'

A few options: "Next week" makes sense if today is Thursday or Friday. "This weekend" would work if today is Monday, Tuesday, or Wednesday (so the day you mean would be Friday, Saturday, or Sunday). You could also say "in a few days" or "in four days." Likely a native speaker would just say the name of the day in question. I don't know of a specific word that means in "the day four days from today."