I agree with bayard on this. However, of the idiomatic forms built on the word good, few make sense. AHD has them as:
as good as Practically; nearly: as good as new.
no good Informal 1. Worthless. 2. Futile; useless: It’s no good arguing with them.
to the good 1. For the best; advantageous. 2. In an advantageous financial position: ended up to the good.
but good Informal Used as an intensive: The pipe started to leak but good.
good and Informal Very; thoroughly: I’ll do it when I’m good and ready.
for good Permanently; forever: I’m moving to Europe for good.
The first three (I’ve regrouped them) make a kind of literal sense. The last three (including the phrase under consideration here) do not.
Certain not-for-profit agencies in my Nothern US rust-belt city are using the phrase “In the City for Good” playing on its literal and idiomatic readings. We’re in the city doing good and we’re not leaving.