Can Park Tool CG-2.2 kit work with the bike upside down?
Yes.

All those tools require is that the chain moves through them. They do require the chain move through in a particular direction so you will probably find that you end up with the handle on the bike side of the chain rather than on the outside, which will be mildly inconvenient. But it beats not being able to clean your chain at all.

Here's what I do, I took the handle off and set it up so the rear derailleur is pulled forward far enough that the chain cleaner can rest on the mech rather than hitting the idler. That lets me pedal with one hand and have a hand free for wiping the chain, chainrings, idlers etc. I also find disposable gloves essential for this, there's no chain lube that's good for your skin.



You could do the same thing with the bike upside down, but you'd need to rearrange things a little.

The question body makes it clear but the original title didn't, so just in case it's not obvious, if you have the tool upside down it will not work - there's a reservoir full of fluid at the bottom, if you invert it that won't hold the fluid.

(note that the above photo is a bit of a cheat, it's of a 406 wheel bike with hub gears so the derailleur is used as a chain tensioner and the chain only runs over one idler. But I do the same thing with my 622 wheeled road bike)