The most common and most likely answer to this crossword clue is the 11 plus 4 letter word FIRST-CLASS POST
I've seen this crossword clue in The Guardian
F
I
R
S
T
-
C
L
A
S
S
P
O
S
T
Buddy explains!
The straight part
We know that "should arrive the next day" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case should arrive the next day = first-class post.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.