Information-theoretic measures for $nl~(^2L)$ states of H atom (with $n=1-10$ and $l=0-2$, where $n$ and $l$ denote principal and angular momentum quantum numbers) have been investigated within a quantum dot by utilizing the Ritz variational principle, with the help of a Slater-type basis set. A well-established two-parameter (depth and width) model finite oscillator potential is used to simulate the dot environment. Variationally optimized position ($r$)-space wave function is utilized to determine the momentum ($p$)-space wave function, leading to the generation of $p$-space radial density distribution. We have explored the impact of cavity parameters on quantum information theoretic measures, such as Shannon ($S$) and Fisher information ($I$) entropy, in ground as well as excited states. The results of $S$ were also used to test the Bialynicki-Birula-Mycielski (BBM) inequality, related to the entropic uncertainty principle for the confined H atom. Some simple new fitting laws pertaining to $S$ and $I$ have been proposed. Furthermore, the $p$-space radial density is employed to derive the Compton profile of the confined H atom. Possible tunability of $S, I$ and Compton profiles with respect to the parameters are noted.