While I did that in the past, I am curious. Can You get this achievement through Arcade?
Or do You have to start a new campaign, in order to get it?

you would need to start a new campaign.

I had to beat the game just to unlock Nightmare now to restart for Ultra-Nightmare.

I wonder if mission select would work if you select the last mission.

(Update) I found out you only need to beat the UAC first level in Ultra-Nightmare to unlock but, I notice you unlock it.