luceos this should have been the process:
install flarum inside your public_html
move all files one level up
rename public to public_html
@luceos is this still relevant?
EDIT: To answer my own question, it's till relevant.
IMPORTANT, if someone come later to this, it's important to change site.php file, this line:
'public' => __DIR__.'/public'
'public' => __DIR__.'/public_html'
And, of course, in config.php point to root domain