From a decision-theory perspective, I should essentially just ignore the possibility that I'm in the first 100 rooms - right?

Well, what do you mean by "essentially ignore"? If you're asking if I should assign substantial credence to the possibility, then yeah, I'd agree. If you're asking whether I should assign literally zero credence to the possibility, so that there are no possible odds -- no matter how ridiculously skewed -- I would accept to bet that I am in one of those rooms... well, now I'm no longer sure. I don't exactly know how to go about setting my credences in the world you describe, but I'm pretty sure assigning 0 probability to every single room isn't it.

Consider this: Let's say you're born in this universe. A short while after you're born, you discover a note in your room saying, "This is room number 37". Do you believe you should update your belief set to favor the hypothesis that you're in room 37 over any other number? If you do, it implies that your prior for the belief that you're in one of the first 100 rooms could not have been 0.

(But. on the other hand, if you think you should update in favor of being in room x when you encounter a note saying "You are in room x", no matter what the value of x, then you aren't probabilistically coherent. So ultimately, I don't think intuition-mongering is very helpful in these exotic scenarios. Consider my room 37 example as an attempt to deconstruct your initial intuition, rather than as an attempt to replace it with some other intuition.)

Theoretically there are as many multiples of 10 as not (both being equinumerous to the integers), but if we define rationality as the "art of winning", then shouldn't I guess "not in a multiple of 10"?

Perhaps, but reproducing this result doesn't require that we consider every room equally likely. For instance, a distribution that attaches a probability of 2^(-n) to being in room n will also tell you to guess that you're not in a multiple of 10. And it has the added advantage of being a possible distribution. It has the apparent disadvantage of arbitrarily privileging smaller numbered rooms, but in the kind of situation you describe, some such arbitrary privileging is unavoidable if you want your beliefs to respect the Kolmogorov axioms.