Rimefire Plate would only prevent 1 damage from Rimehound Bite.
This is really about damage packets, not about resistance. Each time you take damage, Rimefire Plate reduces the damage you take by 1. The real question, then is whether Rimehound Bite deals damage once or twice.
From the RC, p222-223:
When most attacks deal damage, they do so through a damage roll: a roll of dice to determine damage. Whenever a power or other effect requires a damage roll, it specifies which dice to roll and how many of them. For instance, an attack might indicate that it deals 2d8+4 damage on a hit. When a creature hits with that attack, roll 2 eight-sided dice and add 4 to determine how much damage it deals.
Modifiers to Damage Rolls
Many powers, feats, and other game features grant bonuses or penalties to damage rolls. A bonus to a damage roll is added to the damage roll as a whole, not to each die within it. ... If a creature has a bonus to damage rolls and uses such a power, the creature applies the bonus to every damage roll of that power.
Many powers and other effects grant the ability to deal extra damage. Extra damage is always in addition to other damage and is of the same type or types as that damage, unless otherwise noted. ...
Example: Valenae the cleric might have an ability that causes her to deal 5 extra radiant damage to undead creatures. That ability means she deals 5 extra radiant damage whenever she deals damage to an undead creature.
Rimehound Bite's untyped damage and cold damage are both part of the same damage roll. More specifically, the 1d6 cold damage is extra damage, a modifier to the main 1d12 damage roll, just like the +6 is, otherwise it would also receive a +6. Because the damage is all a single damage roll, resistances and vulnerabilities are only applied to it once, so Rimefire Plate's resist all 1 only triggers once.
For example, if you roll a 6 on the d12 and a 3 on the d6, Rimehound Bite would deal 15 damage to a target with no resists, 14 damage to a target in Rimefire Plate, and 12 damage to a target with resist cold 30. The attack would deal 12 untyped damage and 3 cold damage, but because it's all part of the same damage roll resistances and vulnerabilities are only applied once, to the full 15 damage.
RAW vs RAI
For those who aren't rules lawyers, RAI is Rules As Intended, how (we think) the designers meant the rules to work, while RAW is Rules As Written, how the rules (as worded in the book / PDF / clandestine radio transmission) actually work. Generally speaking, it's better to stick with RAW rather than attempt to read the designer's minds to determine RAI, especially since RAW will usually accurately reflect RAI (assuming the designers are competent speakers of whatever language the book is in). Sometimes, unfortunately, the RAW is vague and could reasonably be interpreted in multiple ways (as is the case here), and on rare occasions the RAW is just plain silly, and clearly not RAI. In these cases, it becomes necessary for us to attempt feats of psionic prowess to determine what the RAI really is. Or we can, y'know, be logical about it.
So why should we suspect that the RAI is in fact for each damage roll to be treated as a single instance of damage, and have resistances (particularly resist all) applied to it only once, even if it contains multiple types of damage?
It's simple. It follows the 4e convention of having general rules that can be overridden by specific game elements as needed. It's a lot easier to have the rules say, "Resistances apply to each damage roll once," than it is to talk about how they separately apply to & interact with each different type of damage involved, and if there's a case where resistances really do need to apply separately to each damage type, that game element can explicitly say so.
It's less work. What if an attack deals 3 different kinds of typed damage? Which is easier, applying resist all once to the total, or figuring out how resist all combines with other resists for each type of damage involved?
It makes resist all less overpowered (and resist all is pretty darn strong already). Consider an optimized epic-tier ranger, a firesoul & stormsoul genasi with the Extra Manifestation, Double Manifestation, and Shocking Flame feats, plus the Radiant One epic destiny. Each melee attack with CA deals a certain amount of untyped damage (the normal weapon damage from the power), plus some fire damage (from Shocking Flame), some lightning damage (again from Shocking Flame), and some fire and radiant damage (from Radiant One). Should resist all 5 reduce the damage from this attack by 5? Or by 20?
It's how most people interpret it already. Just ask the official WotC 4e Character Optimization sub-forum, where adding more damage & more damage types is practically the national pastime. Or better yet, ask the official WotC 4e Rules Q&A sub-forum.