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on the go is the same as on going?
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Not really:

'On the go' refers to 'someone on the go.' - Bob is on the go..

'On going' refers to 'something in the process (like an investigation that's on going..)
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"ongoing" - one word - is an adjective.

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Hi,

In the situation when you wonder whether a person had a working holiday in other countries which one would be correct?

(1) Did you go there "on" working holiday? => like in, are you on working holiday?

(2) Did you go there "as" working holiday?