What do ‘it’ ‘on’ ‘off’ mean in “it’ll be on with the old Invisibility Cloak and off to find out what he’s –“

Solution 1:

In this context, on means to put on, and off means will be off and away, while the word it is used as a dummy pronoun rather than as a referential pronoun.

Thus, in the phrase "...it’ll be on with the old Invisibility Cloak and off to find out...", it is a dummy subject, such that "it’ll be on" stands in for "what I will do is to put on".

The upshot: "... the moment I see him lurking somewhere [...] I’ll put on [cloak] and be off and away to find out what he’s [up to]".

Solution 2:

I haven't read the book and I don't follow the series, so I have no idea what Crabbe, Malfoy and Goyle are, but this is how I parse the sentence:

1) It isn't a person, it's generally what will happen, the future, the universe. Like saying, "it's going to be good" or "it's the only solution."

2A) On stands for "put on the Invisibility Cloak"

2B) On can alternatively mean "to continue with the Invisibility Cloak." This is indeterminable out of context.

3) Off stands for "depart on a journey". Therefore:

Original sentence:

And the moment I see him lurking somewhere with Crabbe and Goyle keeping watch outside, it’ll be on with the old Invisibility Cloak and off to find out what he’s ...

Simplified by erasing the first sentence:

And then it’ll be on with the old Invisibility Cloak and off to find out what he’s ...

Explained:

And then the old Invisibility Cloak will be put on and I'm starting the journey to find out what he’s ...