What is the ruling on the acceptance of supplication for one who wakes up after Fajr, or wakes up at night but did not sleep in a state of ritual purity (wudu'), by analogy with the Hadith which states that the supplication of one who sleeps in a state of ritual purity and wakes up at night is answered?

Ablution before sleep is a confirmed Sunnah (definition), based on the hadith (definition) of Al-Bara' ibn 'Azib found in the Two Sahihs and other collections. In Sahih (definition) Muslim, it is narrated from Jabir that the Prophet (peace be upon him) said: "Indeed, there is an hour during the night in which no Muslim man asks Allah for good concerning any matter of this world or the Hereafter but that He gives it to him, and that is every night." The apparent meaning of the hadith is that this is general, and it does not state that the answer is conditional upon sleeping with ablution. The hadith mentioned in the question has not been found, and even if it were authentic, it would not apply to someone who did not perform ablution or who performed it but did not wake up during the night.