"How long after a petition gets approved do you have to pay the NVC?"
Once a petition is approved by the USCIS, you’ll get an invoice from the NVC by email. Once you get it, you have up to one year to pay.
Note that the approval notice from USCIS is only valid for four months for Form 129-F, even though the government will keep your case open for up to 12 months as long as you’re pursuing the visa. However, the I-130 doesn’t expire but can be revoked under certain circumstances.
Keep in mind if there are any discrepancies, USCIS may deny your application, because the validity period of the approval notice has expired. It is therefore recommended that you move forward as soon as you can.