All perfect praise be to Allah, The Lord of The Worlds, and may His peace and blessings be upon our Prophet Muhammad and upon all of his family and companions.
The aforementioned Hadith isn't sound nor was mentioned in authenticated books of Sunnah. Further, jurists labeled it as a fabricated narration and prohibited narrating or circulating it.
In this regard, Al-Imam Al-Shokany said in his book [Al-Foa'eed Al-Majmo'aa] :" The so called narration:( Whoever prays at the last Friday of Ramadan the five obligatory prayers at their times, consequently; all of his missed prayers in that year is will be compensated." .
"I haven't found it even in the fabricated books of Hadith where fake narrations are gathered. However, this narration was circulated by a certain group in Sana recently and by a lot of people although it is fake."
Therefore, whoever missed obligatory prayers and couldn`t figure out their exact number should do his best in that regard. If he couldn't, then he should make up the number he is certain about little by little. This even if he has to offer make up prayers with each obligatory prayer he performs from now on since by time he will manage to make up what he missed, if Allah wills.
Finally, he must repent and make up prayers that he had missed, and performing one prayer as a compensation for missed prayers won't suffice since this whole idea wasn't mentioned in authenticated books of Sunnah nor had been approved by jurists. And Allah the Almighty knows best.