You can post to a page with an Page Access Token, as described in the docs here: https://developers.facebook.com/docs/graph-api/reference/v2.0/page/feed/#publish
A page access token with
publish_actions
permission can be used to publish new posts on behalf of that page.
You can get the Page Access Token by issuing
GET /me/accounts
with the publish_actions
permission (see docs here: https://developers.facebook.com/docs/facebook-login/access-tokens/). Then, exchange the short-lived token to a long lived one:
GET /oauth/access_token?
grant_type=fb_exchange_token&
client_id={app-id}&
client_secret={app-secret}&
fb_exchange_token={short-lived-token}
as described here: https://developers.facebook.com/docs/facebook-login/access-tokens/#extending