The most common and most likely answer to this crossword clue is the 5 letter word FOGEY
I've seen this crossword clue in The Guardian
F
O
G
E
Y
More clues leading to the result FOGEY
Buddy explains!

The straight part
We know that "old conservative" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case old conservative = fogey.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 FOGEY mean?
FOGEY noun- A dull old fellow; a person behind the times, over-conservative, or slow.