I currently have a domain that appears at number one for a not too popular search term (BUSY Schools) In Australia and want to know if I put in place a 301 Redirect to a newer domain that is associated with the same company that is just one month old, if this will take its place at number one.
It’s a new business called “Busy Schools”.
Google rank isn’t a fixed thing. It changes depending on the location of the person searching, their search history and other factors. Search results are also largely dependent on the content of the page, and its calculated relevance to the search term.
If the new page has the exact same content as the old page, then it would probably rank pretty highly. If not … 
Basically 301 Redirects are used to permanently redirect all pages on your old site to your new site which tells search engines and users that your site has permanently moved. Thus it is crucial for your presence on search engines, and your potential incoming traffic.
Permanent Redirection (301) is recommended in your case because it passes 90-99% of link juice (ranking power) to the redirected page.
Cheers!