I want to set a bounty on my question within one day of asking it. I did not see an option for that. Is it possible? If so, how?
You must wait 2 days before you can offer a bounty on your question. You can also edit your question to bump it to the top of the front page.
See the FAQ
If your idea is that people will answer it faster if it is only one day, I think it doesn't work this way.
There generally an advantage to answering a question early, you will get the votes and be at the top (You get momentum). So people probably don't think, oh, I will answer that tomorrow. By leaving it up longer, you will probably just get the best answer over time, and maybe good ones right away.
I covered this topic extensively here
I'll summarize it here.
Without bounties in the first 48 hours, we have a StackOverflow economy that is based on providing timely, quality, relevant answers to questions.
Downsides to immediate bounties
There is no urgent tag
The Purpose Of Bounty
Bounties were created as a method to attract attention to questions which were missed on the initial run-through.
In the words of the FAQ
[Even] the FAQ says that "despite your best efforts". hat means if you've waited a few hours with no response, you can't just buy your way out of it. You should still be making an honest attempt to help the site by improving your question.
Bounties do not improve questions
Please read the original answer for full explanation of these points.
Bounty questions cannot be closed or migrate by normal users, so it would be difficult to process when someone wrongly set bounty when it shouldn't be.
This would require the bounty system to be restructured. It isn't a simple change. People would have to make decisions and implement them.
The problem is that a bounty question has a special status. For example, I'm a 10K+ SO user, and I can't vote to close a question with a bounty. That seems fair, but there are questions out there that really shouldn't be, that we like to close. The current rule is that if the question survives for two days, it's probably OK. (Yes, this rule fails from time to time.)
So, what should happen to a question with a bounty when it's voted closed? Can the questioner still accept an existing answer? Is the rep just refunded? What happens if the question gets deleted? These are questions we'd have to answer to allow immediate bounties.
In the meantime, I'd suggest looking at your question to see why it hasn't been attracting answers.