You want to know?
Okay, I'll tell you.
You inherit it the same way you inherit anything else. You have to be a god, you have to be assigned to the position, and you gradually grow into the panoply and power that the prestige of your positions posits you assuming. Ahlat was promoted to the position of Southern God of War and Cattle, and gradually assumed a muscular, warlike countenance. He was good at his job.
You have to be promoted to the position, though, presumably by a superior - and who is superior to the Most High? You also have to be a god, which rules out mortals, Dragon Kings, and Solars. Daystar, likewise, wouldn't really respond to your Essence and start channeling it (re: start the sun glowing again) until you were truly the Most High, rather than an upstart. You can be named the Most High, but that doesn't immediately make you so. Truly ascending to the might, mindset, righteousness and power of the Unconquered Sun might take years, or centuries, or never. You just might not be good at the job.
But that's how you do it.
There is, however, another way.
Bonus Comment by Nephilpal:
Neall's answer is a half answer.
If a God were appointed a new Sun through a bureaucratic process, the experience would gradually change him. He would become more sun-like, certainly, and more powerful, but he would build on whatever god he was before rather than just inheriting his predecessor's panoply. Outright transference of the mantle as described in Glories is a miracle of significantly higher-order significance. If you want to get an effective but different sun god after he's had a few millennia to grow into the job, then cool. If you need a Sun 2.0 with the full panoply RIGHT NOW, you need a miracle that boggles the imagination of the divine.
Right, I forgot about mentioning the panoply. It's worth noting that the Sun didn't develop his panoply, as it was, until later in his operational lifespan. Anyone who inherits the Sun's mantle through bureaucracy would eventually develop equally-powerful but likely different tools with which to do the job.
If you need a Sun 2.0 with the full panoply RIGHT NOW, you need a miracle that boggles the imagination of the divine.
Yeah, you'd need something that could strum the threads of the Loom itself.
Willing Assumption of Chains would do the job, but while the least (greatest?) god would become more sun-like, they would still need to buy up their Attributes and Abilities. You'd also need to levy it onto an Essence 9 god to start, which means you'd need to buff Ahlat or Shining Flower or Tachi-Kun long beforehand.
An Essence 10 deity who is not an Incarna could be Willed to the position of Most High, but even they would take years or decades to start really developing into a proper Sun and get Daystar's light bright again.