As a PR, you must be residing in Canada in order to be eligible to sponsor your dependent. Only Canadian citizens are eligible to sponsor family while they're living abroad.
There is no inland sponsorship for dependent children - the application would be processed through Buffalo. That doesn't mean that your child definitely has to stay in the States, though, during processing. Because s/he is visa exempt, it's possible to be admitted to Canada as a visitor to wait with you for the application to be finalized. Dependent child sponsorships processed through Buffalo typically take from 2-8 months, plus another 38 days or so at the Case Processing Centre in Mississauga where they assess you as an eligible sponsor.
You should start working because you'll need to demonstrate that you've settled in Canada in order to be eligible to sponsor. There is no minimum income requirement to sponsor a dependent, but you'll need to demonstrate that you can support your child and not have to apply for social assistance.