You can post things on Product Hunt if you have an account that is a "contributor" [1]. When I did it, it meant fulfilling some easy requirements like upvoting items on the site for three days in a row.
Quite new to Product Hunt myself but I believe you can publish/launch the product yourself or for more impact, you can reach out to some of the top hunters to see if they want to collaborate.
the featured tab is manually curated. If they like your submission it's possible to end up on the front page the day after you submit, even with few votes (this happened to me a few times).