Checking the mysql error would have revealed:
You have an error in your SQL syntax; check the manual that
corresponds to your MySQL server version for the right syntax to use
near '= 'AT275981020' then '700' end case'
I don't know if you thought you could write case as an alias or what that alias could be used for. Maybe this was just bad copy-pasta from another CASE block that you found in a SELECT query.
Ultimately it was the trailing case that was the issue. The query will execute once that final word is removed, but there is more to explain.
In my own project, I would be using the more concise variant of CASE syntax with ID only written once instead of writing mostly redundant expressions in each case.
I also recommend using lowercase column names so that the sql is easier to read for humans.
UPDATE ae44 SET
Price = CASE ID
WHEN 'AT259793380' THEN '500'
WHEN 'AT271729590' THEN '600'
WHEN 'AT275981020' THEN '700'
END
WHERE
ID IN ('AT259793380','AT271729590','AT275981020')
With the addition of the WHERE cause, only 3 rows will be whitelisted for possible modification -- as a consequence the ELSE can be safely omitted from the CASE block.