The most common and most likely answer to this crossword clue is the 5 plus 3 plus 5 letter word ALPHA AND OMEGA
I've seen this crossword clue in The Guardian
A
L
P
H
A
A
N
D
O
M
E
G
A
Buddy explains!
The straight part
We know that "Characters in old script from beginning also finish" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case characters in old script from beginning also finish = alpha and omega.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.