The most common and most likely answer to this crossword clue is the 4 letter word STIR
I've seen this crossword clue in The Daily Telegraph
S
T
I
R
Buddy explains!

The straight part
We know that "Wake up/cooler" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case wake up/cooler = stir.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 STIR mean?
STIR noun- The act or result of stirring (moving around the particles of a liquid etc.)
- Agitation; tumult; bustle; noise or various movements.
- Public disturbance or commotion; tumultuous disorder; seditious uproar.
- Agitation of thoughts; conflicting passions.
- To incite to action
- To disturb the relative position of the particles of, as of a liquid, by passing something through it; to agitate.
- To agitate the content of (a container), by passing something through it.
- To bring into debate; to agitate; to moot.
- To change the place of in any manner; to move.
- To move; to change one’s position.
- To be in motion; to be active or bustling; to exert or busy oneself.
- To become the object of notice; to be on foot.
- To rise, or be up and about, in the morning.
- Jail; prison.