Given a set X with a certain cardinality, there are explicit constructions for getting a set with the "next bigger" cardinality, e.g. constructing the power set.
Does some analogous construction exist for the reverse, i.e. getting a set with the next smaller cardinality?
In general no, because not every cardinality has a "next-smaller" cardinality. In particular, if $\kappa$ is a limit cardinal then it is not a successor cardinal, so there is no cardinal $\lambda$ such that $\kappa = \lambda^+$.
However, if $\kappa$ is a successor cardinal, or if for a limit cardinal you're ok with finding a subset of any smaller cardinality, the procedure is the same:
Let $\kappa$ well-order $S$, pick your desired cardinal $\lambda < \kappa$, let $x$ be the element of $S$ with order-type $\lambda$, and then let $T = \{s \in S : s < x\}$.