The App Access Token is the most basic one, and will not allow you to post anything. In order to post something to a Facebook Page (as a Page), you need to get a Page Access Token.
The process is a bit complicated, because you actually need to authorize the user with the "manage_pages" permission first, with the User Access Token you can call the API to get a Page Access Token (/me/accounts).
See those links:
- https://developers.facebook.com/docs/facebook-login/
- https://developers.facebook.com/docs/facebook-login/access-tokens/
- http://www.devils-heaven.com/facebook-access-tokens/
Btw, the REST API is deprecated: https://developers.facebook.com/blog/post/616/
You can also try the "Client Token" (Developer Settings > Advanced), i never worked with that one but it could be the easiest solution. In any case, an App Access Token is the wrong one.