Suppose I have a question that is already made, but it has been there from a long time and without no answer. If I want to bring that question to the top as every new question by users, the only way I can put that question again at the begining of the queue is by adding a bounty on it? Shouldn't be another way to do that?, I know that even if it is old there are choices for it to have an answer, but it's not the same as if it had been placed at the begining of the queue once again.
If you have enough rep, offer a bounty.
If you don't have enough rep, honestly, just re-ask the question. It will either get closed as a duplicate, drawing attention to the original question, or it will get answered. I'm not sure if this is the best way to go about it, but it is a way. And I don't think anyone can fault you for re-asking.
As always vote up for agree or down for disagree.
A while ago I downvoted the existing answer that recommends re-asking. Now I encountered a concrete case (Covariance of 2 Shortest Pieces in Broken Stick), where I asked someone not to re-ask questions, and then when they asked what else to do I had to admit that the only answer in this thread (currently with $14$ upvotes and $4$ downvotes) suggests to re-ask. So I figured I need to write another answer :-)
Two things need to be balanced here. On the one hand, we don't want duplicate questions – even if you link them, it becomes harder to keep track of progress and compare answers, and it's one more question that fills up search space etc. On the other hand, the person now asking the question should be able to ask their question; we can't just tell them, tough luck, it's already been asked. I've been focussing on answering old unanswered questions recently, so I know there are lots that could be answered but haven't. (The question mentioned above was such a case.)
One approach would be to improve the existing question. Often questions aren't answered because they're badly phrased and/or the OP doesn't react to requests for clarification. You could revamp the question, clarify it and make it more attractive to answer. That would bump it onto the active questions list, and you'd be contributing to the site in the process.
In the present case, that wouldn't have been a good option, since the existing question presented some partial work and asked about its correctness. When a bounty and an improvement of the existing question aren't viable options, it could be OK to ask a new question; but I think in this case the requirement to include your work weighs particularly heavily. You should try to digest everything that was said about the problem in the existing thread (which was quite a bit in this case), try to build on it, and indicate where and why you can't make any further progress (or perhaps where previous attempts went wrong or are unhelpful). In this way, you'd be making a contribution to the site, and instead of just duplicating a question you'd be asking it with a new angle and contributing to answering it.
Of course sometimes one just doesn't have any idea how to approach a problem. But the probability that neither a bounty, nor an improvement in the existing question, nor a minimal contribution towards answering the question are possible seems quite low. If so, if you really must simply duplicate the question, please link the questions in both directions (in the text of your own question and in a comment under the existing question) and vote to close one as the duplicate of the other as soon as possible, when an answer has been provided and upvoted or accepted. (This of course also applies if you re-ask it with a new angle.)