There's no predictable system, and it varies on each route. Best is to set an alert on ExpertFlyer ($10 a month subscription, good value) and check regularly. It might open up, it might now. Depends on whether EK sells (or thinks they can sell) F seats.

If you're flexible you may want to check availability on earlier/later flights and see if EK can switch you and upgrade you in one phone call and maybe waive the change fee if you ask nicely.