Error converting data type bigint to varchar.
You can't concatenate a string to a number. You have to convert it:
SET RegistrationFee = 'fee_' + LTRIM(STR(@ID))
You need to explicitly convert your bigint to varchar:
DECLARE @ID BIGINT
set @ID = 1323
UPDATE School
SET RegistrationFee = 'fee_' + CAST(@ID AS VARCHAR(15))
WHERE SchoolRegistrationId = 123
T-SQL will not do this automatically for you - you need to be explicit and clear about it.