How to rank one keyword on first page of google in one month?

There are many thing you can do to make a keywords on top in a month. try these it might help you:
1 Tray PPC
2 Create Authority links for ex create blogs, infographic and articles and try to make hyperlink on particular url with relevant landing page
3 Done blog commenting on high PR Sites for the content you have created it increases particular page session
4 Do Content outreach and try to get links from niche sites

Use pay per click and ensure you place a higher bid than your competitors..can be the fastest way

Maintain its Onpage and do offpage activities for that Keyword.

Quality back links and well optimize websites with daily good content blog updates

main point is domain age

updated content with good back link

optimized website with quality content, back links, daily blog section updates

Ads is the fastest way but it will cost you a lot and it is time limited. Once you stop paid, it drops. Step by step, you need to do some quiality backlinks.

... And the article content is also important.

Quality backlinks and quality traffic

You need to get a lot of back links.