Is it permissible to follow a gradual approach with an alcoholic who wishes to stop drinking, by increasing supplication and refraining from prayer while intoxicated?

The questioner will receive a great reward for his endeavor in guiding people. The caller to Islam is advised to use wisdom and good exhortation, and to explain the goodness of adhering to the truth and abandoning falsehood, and the danger of praying while intoxicated. The caller should show his eagerness and love for the good of others with a gentle and compassionate approach, as the Prophet (peace and blessings be upon him) enjoined gentleness and facilitation.

If gradualism in da'wah means using wisdom by first encouraging the addressed person, then clarifying the Sharia ruling, and observing the appropriate time, then there is no harm in that.

Finally, guidance is in the hands of Allah alone. The caller to Islam must take the right means, not despair, seek help from Allah, and supplicate for the guidance of the addressed person, as the Prophet (peace and blessings be upon him) did with the Daws tribe.