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.
If alcohol has made him drowsy and he felt that he has begun to talk incoherently, then he is not allowed to join prayer in that condition until drunkenness goes away from him for Allaah The Almighty says (what means): {Do not draw near to prayer (when) you are drunken.}[Quran 4:43]
Al-Qurtubi said in his Tafseer (interpretation of the Quran): "If one does not know what he is saying, he should keep away from the Masjid (mosque) lest he pollutes it and his prayer is not valid; and if he offers prayer he has to make up for it. However, if he knows what he is saying and offers prayer, then he will be deemed like someone who is sober (not drunk)."
Shaykh Al-Islam Ibn Taymiyyah said: "The prayer of the drunk who does not know what he is saying is not permissible according to the consensus of the scholars. Rather, it is not permissible to enable him to enter the Masjid for that verse (Quran 4:43) and other pieces of evidence. That is because the prohibition is for drawing near to prayer and the places of prayer..."
Hence, it is incumbent upon that person, along with repenting to Allaah The Almighty, to make up the missed prayers after he becomes sober. It was stated in Asna Al-Mataalib (a Shaafi'i book): "It is obligatory upon him to make up for them (the missed prayers) when he is no longer drunk..."
The same thing was stated in Mataalib 'Uli An-Nuha (a Hanbali book): "The drunken has to make up the prayers he missed during his drunkenness..."
For more benefit about the prohibition and punishment of drinking alcohol, kindly refer to Fataawa 86328 and 115699.
Allaah Knows best.