What must one who broke their fast for two days in Ramadan due to masturbation do to make up for their fasting, and does the hadith "Whoever breaks a day of fasting in Ramadan, a lifetime of fasting will not make up for it, even if he fasts it" mean that there is no repentance for him?

You must repent to Allah and make up for the two days you broke your fast due to masturbation. There is no expiation (kaffarah) upon you. As for the hadith mentioned, there is disagreement among scholars regarding its authenticity, with some deeming it weak and others authentic. Even if it is authentic, what is meant by it is the inability to fully attain the complete virtue of performing an act on time when making it up, not that the obligation to make it up is waived. Sincere repentance is accepted for any sin.