Callous Oppressor enters, naming Cephalid. Next upkeep, Cryptoplasm copies Callous Oppressor.

706.7a says: … If an ability refers to an undefined choice, that part of the ability won't do anything.
In this case, which part isn't doing anything? Does the ability now look like this "{T}: Gain control of target creature that isn't of the chosen type for as long as Callous Oppressor remains tapped."

What can Cryptoplasm's new activated ability do? Can it gain control of any creature? If not, can it target a Phantasmal Image to trigger its sacrifice ability, or tap at all?

The ability Cryptoplasm copied from Callous Oppressor cannot be legally activated.

Callous Oppressor's abilities are linked; the activated ability refers to the choice made by the second ability. No such choice was made for Cryptoplasm, so the choice is “undefined”, and if an ability refers to an undefined choice, that part of the ability doesn't do anything. (CR706.7a)

In the case of Callous Oppressor, the part of the ability that doesn't do anything is the targeting restriction, which means no legal target can be chosen, so the ability therefore cannot be activated.