All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
It is well-known that it is permissible to act in accordance with a Dha‘eef Hadeeth in matters pertaining to virtuous deeds as long as it is not extremely weak. Imaam An-Nawawi said, summarizing the standard rule on using weak Ahaadeeth: "Scholars agreed that a weak Hadeeth may be used in matters pertaining to virtuous deeds, but may not be used as primary evidence in determining what is Halaal and Haraam (lawful and unlawful)." [Al-Majmoo’]
Al-Khateeb Ash-Shirbeeni wrote: "The conditions for acting in accordance with a weak Hadeeth in relation to virtuous deeds are that: it should not be extremely weak; it should fall under a general religious principle already present (i.e. should have basis in the Sharee‘ah); and the one acting in accordance with it should not believe that the action is part of Sunnah (i.e. should not attribute it to the Prophet, sallallaahu ‘alayhi wa sallam) .” [Mughni Al-Muhtaaj]
For more benefit, please refer to Fataawa 13202 and 166199.
Therefore, you are advised to memorize the Ahaadeeth in this book, even the weak ones. There is no harm in memorizing such Ahaadeeth or in acting in accordance with weak Ahaadeeth that the (above-named) editor has not described as extremely weak.
Allaah Knows best.