The help page for bounties says:
The bounty period lasts 7 days. Bounties must have a minimum duration of at least 1 day.
Does this mean I can manually award a bounty after 24 hours have elapsed?
Yes. One day after the bounty is started, you may award it to any answer you choose by clicking the blue bounty number (e.g. "+100") next to it. You may not award the bounty until that full day has elapsed. The age of the answer receiving the bounty does not matter.
Further reading: How does the bounty system work?