Is it permissible to access an ex-fiancé's Facebook page to find out his news, hoping that might help in forgetting him?

There is no harm in visiting a young man's Facebook page for legitimate purposes, provided one is safe from fitna. It is permissible to propose marriage to him with Islamic etiquette.

If marriage is not facilitated, there are ways to treat attachment to an fiancé after separation.

You must increase your supplications and ask for a righteous husband, and exert efforts to search for him by seeking help from relatives and female friends.

Strive to fill your free time with what benefits you in this world and the hereafter. Be diligent in fasting, seeking beneficial knowledge, performing righteous deeds, and keeping company with righteous women. You may also entertain yourself with permissible activities within Islamic guidelines.