Check out page 34 of May-Ponto "More Concise Algebraic Topology" for a more modern treatment of the homotopy limit of a sequence of spaces. In particular, Proposition 2.2.9 is exactly the surjection you're asking about. Their proof is to realize $X$ as a homotopy equalizer, then apply Proposition 2.2.7, which is a general statement about homotopy equalizers. So it seems this proof generalizes completely to the non-countable case. You simply have to realize the homotopy limit of an uncountable sequence as a homotopy equalizer and like May and Ponto we should set $Y = \prod X_\alpha$ and look at the homotopy equalizer of $id_Y$ and $\prod f_\alpha$ where the $f_\alpha:X_{\alpha+1}\to X_\alpha$ are the maps in the system. Incidentally, it's not that uncommon for statements from before 1990 or so to have only been proven for the countable case, but often very similar proofs work far more generally. A good example is Neeman's book on triangulated categories, which introduces the notion of a well-generated triangulated category.

For your second question, on model categories, the reference I use is Hirschhorn Chapters 18-19. Chapter 18 is for simplicial model categories, chapter 19 is in full generality, using framings. For simplicity I'll focus on the material in Chapter 18, but everything has a corresponding generalization in chapter 19. On page 382 Hirschhorn gives the general definition for homotopy limits in model categories. On page 383, right after Theorem 18.1.10 (which is interesting in its own right) he compares his definition to the one in the Bousfield-Kan lecture notes mentioned in your question (that's citation 14 for Hirschhorn), and he points out an error in Bousfield-Kan.

One difference with the classical situation is that instead of working with $[S^n,X]$ one works with the simplicial mapping space. Another is that it's very hard for a model category to say that a map is surjective, but it can say something is a weak equivalence. The theorem you started with doesn't just give the surjection; it identifies the kernel as a $lim^1$ term. So it can be phrased as looking for an isomorphism. The closest thing in Hirschhorn I can find is Theorem 18.7.4 and it's corollaries (which are stated for hocolim but have natural dual versions for holim). There's a lot of stuff in Hirschhorn, so if this doesn't sound like the model category version of the statement for Top it's very possible a different result of his will be a better fit. I encourage people to leave comments if they think there's a better model category analogue.