Ship Pro and Super Pro subscribers have the ability to schedule when their posts go live on Product Hunt. If you'd like to upgrade, you can find more information here.
To do so, you will first go to the Edit page for your Upcoming Page. To get there, you can go here, and then select your Upcoming Page.
After selecting your Upcoming Page, 'Schedule Launch' will be listed at the bottom of the column on the left side of the page.
Select 'Schedule Launch' and you will be directed to the regular post flow with the addition of the option to schedule when you'd like the page to be published.
You can schedule up to 7 days in advance. The page will not be accessible to others prior to the date and time that it is set to be published.
Remember, the time for when the post will be published is in PST.