Last Moment Upgrade to Business class, is it possible?

I'm assuming this is a revenue (paid) ticket, not a mileage award. You say that you booked economy because business was sold out. Since you are willing to buy business class, your travel agency or Etihad should be able to wait-list you now for business class. That way, if someone cancels, your upgrade would automatically clear, your travel agency will be notified, and you can pay the fare difference and have the ticket re-issued. Another option would be to change the day or time of the flight to one where business class is available.

If this is an award ticket, then business class may be full or may have plenty of availability. If the latter, Etihad might release more award seats in days or hours before the flight.

As you seemed willing to pay for business class outright when you originally booked, I would simply keep checking with Etihad's website to see if any business class seats open up. If they do, change your booking.

If you are still in economy when you check in at the airport, ask one more time if any seats have opened up.

Once you are checked in and at the gate, upgrading usually follows the airline's policy, which means highly unlikely unless you hold high elite status and/or a full fare economy ticket. Once in a while airlines will "sell" upgrades at the gate, but that is usually when there are lots of empty BC seats.

The easiest way to do this is to ask for an upgrade at the time of check-in. If it is not available at that time; they will likely put you on a waiting list for an upgrade.

You can confirm your status with the gate agent (where they have a better idea of availability).

Are you likely to get it? It depends on how full the flight is. The airline will try to accommodate you as much as they can (especially as you are a paying customer).

I would not wait and check the website though - the best choice will be making sure you highlight this at the check-in and make yourself known early at the check-in gate.

Arriving earlier than usual would also be advantageous (the wait list is first come, first served).

Airlines work in mysterious ways so this can happen but it's rare. I don't know about Etihad specifically but most airlines have a carefully groomed pecking orders for upgrades. Willing to pay will increase your chances but factors like status and original fare class play a role too. If there are open seats, most airlines will try to sell last minutes upgrades to passengers depending on where they sit in the pecking order. If no one bites for cash the freebies go next based on status and quality of the upgrade units.

I once got upgraded by United (one of the more stingier airlines) on a long haul to China after I already had buckled in to economy. I had no particular status at the time (just silver or gold perhaps) and I did not even applied for an upgrade.

The flight seemed reasonably empty at check in but all of a sudden a huge crowd of Chinese youngsters carrying classical instruments piled in taking every available seat. My guess is that they oversold E and E+ at the last minute and needed and extra seat to accommodate the whole group. I just happened to be lucky enough to be the ranking E customers on this flight and so I get to sit in the front. Happened just once and never again even though I have much higher status now.