Here (in the context of Abstract Elementary Classes) on the page 43 at the bottom,-6th line, what does it technically mean $$\leq_{\frak K_\lambda}-\text{increasing continuous}$$
? I think that this should be a condition on limit ordinals, but in his text, Shelah uses $\alpha$ for both, limit and successors ordinals (see the page 67 in the link above) and he writes in that -6th line
[...for] $\alpha<\lambda^+$.
Note: the relevant definition is at the bottom of page $43$.
You aren't misunderstanding anything, but you are overthinking a bit. Shelah could have indeed written "for all limit $\alpha<\lambda^+$," as you observe, but he didn't need to: continuity is a vacuous condition at successor ordinals. Thinking about it topologically, each successor ordinal $\beta$ is an isolated point, and so there are no restrictions at all on how a continuous function needs to behave at $\beta$.