Ultra-Nightmare difficulty is unlocked after you complete level 6. You have to start a new game to select the setting, you can not use your existing save.
If you're having trouble with this achievement a simpler* approach I would recommend is to employ a speedrun strat.
You'll still need to clear the first batch of enemies but once you're past them its just a matter of getting the jumps correct up the mountainside.
In the video below you can see the approach "DraQu" took to get through the level faster. This also works on Ultra-Nightmare to help net you the achievement.