Depending on where the bounty campaign started, and when I say where, I mean if it is posted here on bitcointalk or it is hosted within a website or bounty platform.
If it is here on bitcointalk, then it is simple, make it a habit to check the first page of the bounty thread; bounty managers often post the announcements and updates on the first post of the first page.
You can also join the bounty telegram group of such bounty campaign to get updates or check the social media channels of the ICO/Project. Bounty managers nowadays usually have their own telegram/discord group for their bounty participants on all of the bounty campaign they handle, you can join those and get updates too, even talk to some bounty managers/admins.