Is attaching a paper to count the days of Ramadan considered a reminder of the passage of life and a righteous deed that has evidence from the righteous predecessors, and is it a form of informing and reminding?

There is no objection to hanging admonitory signs at the mosque entrance, because they serve as a reminder and a means of teaching people. Means take the ruling of their ends, so if the end is good, the means are good. There is a difference between means and innovations (bid'ah); means are not intended for their own sake but serve an objective, whereas an innovator intends the innovation for its own sake. Sheikh Ibn 'Uthaymeen permitted this act, as it is a means to a legitimate matter. Reminding of the remembrance of Allah is permissible, and a means to a legitimate matter is itself legitimate.