From my research, I understand Emirates is very generous in offering every available seat for miles upgrades at airport check-in right through to when the aircraft doors are closed. If a seat is free even when all passengers have boarded, you can seemingly upgrade to it with miles, even on board.
Can I upgrade Emirates after booking?
Most tickets can be upgraded immediately after the booking has been finalised, although the Economy Class Saver fares don't permit this until check-in, with the cheaper Economy Special ones not allowing it at all.
Does Emirates offer Business Class upgrades?
Can you ask for business class upgrade?
Usually just asking a flight attendant for an upgrade won't work, but there are a couple of cases where your chances largely increase. If economy class has been overbooked and there are seats in business, make sure you're quick to offer (this helps when you're sitting in the first couple of rows).
Is it better to book business or upgrade later?
If luxury and extra legroom is really important to you, book it in advance. However, if you're perfectly willing to fly economy class, and willing to roll the dice, a last-minute upgrade can be the cheapest option. You may even get lucky and score a free upgrade.