After the Ring of Lucii update, this achievement is actually a lot easier. It will still take maybe 5-10 mins to complete based on RnG, but it is a lot quiker than fighting the Adamantoise normally which can take over 30 mins.
Basically once you do all the steps to get the Hunt for the Adamantoise, bring a bunch of Magitek Boosters and Elixirs to make this a little easier on yourself.
Pop the Magitek Booster on Noctis to Null out MP costs for using Alterna (R bumper + Y (hold)). You can cast Alterna 3 times before you have to take an Elixir to refill your MP. Then take another Magitek Booster, rinse and repeat.
It took me about 10-20 casts of Alterna before the Adamantoise was instantly killed from full health.
There are some rumors that equipping items that increase your Magic Attack can help, but I think it is complete RnG.