UPDATE: Quick clarification regarding this…the first person mode is accessible following getting a power up. Once you get that power up, you can play in 1st person mode for a limited time (30 seconds, according to leaks).
According to leaked videos from Call of Duty: Black Ops 3, the Dead Ops Arcade 2 mode that is available in-game can be played in either first or third person. This is different from Dead Ops Arcade 1, as that was only available in third person mode.
*Actual in-game images not shown because of Activision take down requests. Image above is of Dead Ops Arcade 1.