In Cloudflare.com once you log in you should see an "Overview" tab, scroll down that page and look for a section that says "Change Plan".
You will then be presented and asked to choose your Cloudflare Plan. Here you will be shown the number of remaining Enterprise Plan slots you have remaining. Select "Enterprise Website" and then select "Continue".
Assuming you have paid for Enterprise, you should see an option to add in the x number of Enterprise Websites. Select that option to enable Enterprise for your domain.
Then choose which websites you want to be upgraded to Enterprise.
Confirm the number of websites that will be upgraded to Enterprise.
Wait and allow up to 24 hours for changes to be processed. As per details specified on the page.
There will be no downtime when you switch your name servers. Traffic will gracefully roll from your old name servers to the new name servers without interruption. Your site will remain available throughout the switch.