If Containment Priest enters the battlefield without being cast, its ability won't exile itself.

Containment Priest's last ability won't affect any creatures that were cast, including ones cast from unusual zones such as your graveyard.

If Containment Priest enters the battlefield at the same time as other creatures, its ability won't affect those creatures.

If a noncreature card wasn't cast and is entering the battlefield as a creature (due to an effect such as that of March of the Machines), it will be exiled. Conversely, if a creature card wasn't cast and is entering the battlefield as a noncreature permanent (for example, Heliod, Sun-Crowned with insufficient devotion), it won't be exiled.