I tried searching for the word 'aroo' which sounds as though it ought to mean something but came up with nothing specific.
The term 'switch-a-roo' leading me to a shortening of 'around' but as I say, not much help.
Seeing as a-rooki-ooky isnt found anywhere else (as far as I know) would lead me to believe that it is a pure made-up nonsense.
Spelling can not be an issue, as the majority of folk in those days did not spel to gud :-)
Maybe the term was a corruption of a foreign phrase picked up in some far flung exotic port like Necastle ......
Okie dokey?