The most common and most likely answer to this crossword clue is the 5 letter word STUPA
S
T
U
P
A
Buddy explains!

The straight part
We know that "shrine" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case shrine = stupa.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does STUPA mean?
STUPA noun- A dome-shaped Buddhist monument, used to house relics of the Lord Buddha.
- A stupe (medicated cloth or sponge).