The most common and most likely answer to this crossword clue is the 5 letter word FUSTY
I've seen this crossword clue in The Times
F
U
S
T
Y
More clues leading to the result FUSTY
Buddy explains!

The straight part
We know that "Old-fashioned" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case old-fashioned = fusty.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 FUSTY mean?
FUSTY adjective- Moldy or musty.
- Stale-smelling or stuffy.
- (by extension) Old-fashioned, refusing to change or update.
- (of wine) Tasting of the cask.