My Deltarune theory is that Rudy is sick the same way Berdly gets sick in the weird route
Berdly is fatally injured in the weird route while in the Dark World when Noelle uses the spell Snowgrave on him. This somehow differentiates from when the other Lighteners (Kris, Susie and Noelle) reach 0 HP, because they are able to bounce back from it and are seemingly unaffected by their Dark World injuries when they return to the Light World. My guess is that Noelle is stronger than the other Lighteners due to the Holidays being a boss monster family similar to the Dreemurrs (confirmed by the Undertale 10 year anniversary event). Put a pin in that, it's important.
Rudy is chronically ill, although the details of his illness are not explicitly stated. He is indefinitely hospitalized, only leaving to attend church and other special occasions such as the chapter 5 festival. His main symptoms seem to be respiratory as he coughs a lot and doesn't seem to have issues with the rest of his body (outside of stamina) because we see him climb a ladder in chapter 5. (Side note, Rudy is stated by Carol to have "fallen down" later in the chapter. This is the same terminology used in Undertale when a monster passes away. However, since we last saw him on a ladder, it is possible she's just referring to him falling from the ladder. The fate of Rudy is left ambiguous at the moment, but the point is that he likely overexerted himself by attending church and the festival on back to back days, leading to him either passing away or literally falling from the ladder)
Chapter 5 confirms that Dark Worlds have existed in Hometown for a long time through Asgore's dialogue. Asgore recognizes certain Darkeners such as Flowery and he states that his obsession with learning more about the Dark Worlds is what caused the Dreemurr household to divorce. We know that the divorce is not recent, as it is implied that his obsession with the Dark World was driven by a desire to find December Holiday. Her disappearance (in an ambiguous event that also traumatized Kris and Noelle) took place early enough for Noelle to have lost the spelling bee to the word "December" as a young child. I'd estimate that to be about 10 years ago since Noelle is near the end of highschool (we know she's near the end because Berdly talks about applying to colleges and they participated in the spelling bee against each other). Anyways the point is that Dark Worlds have been in Hometown for a long time.
Cutting to the chase, I believe that Asgore and Rudy entered a Dark World together years before the start of chapter 1. We know the Holiday and Dreemurr households are very close, so it's not unlikely that when Asgore stumbled across the Dark World for the first time, so did Rudy. We also know that Asgore likely faced the Roaring Knight while in the Dark World, since in chapter 4 we can overhear him talk about a "black shard" which is the same item that can be obtained when defeating the Knight in chapter 3 and breaking off a piece of its sword.
In the world of Deltarune, monsters don't have the same magic skills they do in Undertale with the exception of when they are in the Dark World. In Undertale, Asgore and every other Dreemurr is seen using fire magic, meaning that Deltarune Asgore likely had fire magic when he entered the Dark World. It's the next closest thing to canon information that Asgore engaged in combat while in the Dark World (due both to him having a black shard and also because of the hostile nature of most Dark Worlds (I believe the chapter 3 Dark World is the only one that doesn't immediately try to attack the party)
Holidays, while not directly appearing in Undertale (although canonically there), do also have a family magic trait. It's ice. No shocker there. They're a bunch of reindeer and their last name is Holiday. This part is obvious. (And it confirmed via the Undertale 10 year anniversary event that the Holidays lived in Snowdin). Also we see that Noelle Holiday casts solely ice spells while in the Dark World so that double confirms it. Anyways.
There is a world in which Rudy is caught in a fire spell by Asgore while they are in the Dark World. Perhaps a building catches on fire and he suffers from breathing in the smoke. This would line up with the respiratory issues he develops. And perhaps being caused by a boss monster such as Asgore, the damage lingers when they exit into the Light World just like Noelle's damage to Berdly lingers.
Also, not evidence but a nice touch, Asgore brings Rudy flowers in the hospital in what seems to be a semi-regular occurrence. Perhaps to say sorry for causing the illness? Or perhaps he is just a man worried about his sick friend and wanting to bring a bit of color to the sterile hospital.
Anyways. This was my theory. I'd like to hear your thoughts :)
