I need to modify the apache conf and the .htaccess
No, you don't. One or the other. If you have access to the config file, use it. Anything directory-specific goes in a <Directory> envelope. At most, allow htaccess for trivial things like indexing (overrides Indexes and/or Options). If you don't have access to the config file, you've no choice but to use htaccess.
at the moment the only way to access my site is by typing:
what i need is to access it without typing the account_name:
:: please stop here and detour to the post about using example.com since Our Moderator went to the trouble of writing it and examples won't work with anything else ::
Is yours the only account name in the domain? Did there use to be more of them and you've now got rid of the others? Does the content live inside the /~account_name/ directory? Are there directories other than /public/ ?
So far I can't figure out whether you need a Redirect or a Rewrite, and whether you've got an internal problem or a DNS one.