How do you get to divine beast Rudania in Breath of the Wild without burning to death?
You will need to use heat-resistant armor/elixirs to travel through this area. Most notably, you will want the Flamebreaker armor. There is a side quest you can do (and you really should do) to get this armor. Around Death Mountain Marker #3, you should look for an NPC named Kima. He will give you the side quest that once you complete rewards you with the armor. From Polygon:
The road through Death Mountain has a few markers that allow you to chart your progress. Beyond the third marker, look for a man named Kima running around with an exclamation point above his head. He’ll unlock the Fireproof Lizard Roundup side quest, request 10 fireproof lizards and offer you heat-resistant armor in return.
The easiest way to find fireproof lizards is to snap a picture of them with your camera, and set your Sheikah slate to detect them. Then follow the signals, crouching to grab them. (There are a bunch of fireproof lizards around the Southern Mine.) Collect 10, speak to Kima, and you’ll get Flamebreaker armor.
The side quest itself is pretty straight forward, and the linked article provides an easy way to complete the quest. Note that completing the side quest does not give you the helmet for the armor, which is needed for the set bonus. The only way to get the helmet is to purchase it from the Ripped and Shredded armor shop in Goron City.
Alternatively, if you wish to purchase the armor outright from the armor shop mentioned, it will cost you 3,300 rupees. You will need to make use of elixirs (see below) to make it there first. The idea behind the side quest is if you do it, you get the armor for free, while only needing to spend 2,000 rupees to get the helmet from the armor shop. If you upgrade the Flamebreaker armor to level 2, you will gain the fireproof bonus.
In addition to the armor, perhaps you should look into Fireproof Elixirs. You will need fireproof lizards or Smotherwing butterflys and monster parts to make them. You may also buy Fireproof elixirs from Gaile in Foothill Stables, which is near the maw of Death Mountain.