You Do Need power plug adapter while traveling from Cayman Islands to Cyprus.

Here is Why You Need An Adapter

The outlet and plug type used in Cyprus is G and in Cayman Islands are A and B. This is how they look like.

Cayman Islands

In Cayman Islands, outlets of type A and B are in use.

Type A

Type B

Cyprus

In Cyprus, outlets of type G are in use.

Type G