My wife and I are trying to best utilize our time in the Czech Republic and would like to visit the Prague Castle on Monday, April 23
I suggest taking a 2 1/2 hour Prague Castle walking tour with a knowledgeable tour guide. It'll save you time, and you don't have to wait in line for the entrance tickets. The tour usually includes St. Vitus Cathedral , Starý královský palác, St. George's Basilica and Zlatá ulička u Daliborky 12/1(Golden Lane).
Take a look at Viator.com's Prague Castle Walking Tour and Admission Tickets package: https://www.viator.com/tours/Prague/Prague-Castle-Walking-Tour-Including-Admission-Tickets/d462-9396P1
Prague Castle is open daily. As Mary wrote, there are different buildings inside the castle. During my visit, the St. Vitus Cathedral was closed. Here's a list of days of possible closure dates: https://www.hrad.cz/en/prague-castle-for-visitors/archive-of-closures
Mentioned in this answer:
Prague castle is a complex of different buildings rather than just one 'castle'. The official castle website in English gives the opening days (every day, basically) and times for each: