You Do Need power plug adapter while traveling from Austria to Ireland (Eire).

Here is Why You Need An Adapter

The outlet and plug type used in Ireland (Eire) is G and in Austria are C and F. This is how they look like.

Austria

In Austria, outlets of type C and F are in use.

Type C

Type F

Ireland (Eire)

In Ireland (Eire), outlets of type G are in use.

Type G