I think Miss Carrow may be overly cautious. Anyone may sit the NEWT exam provided an application is received on time. It is generally not advised if one has not had the requisite number of years of instruction. You may be asked to complete a practical OWL, however.

There have been numerous cases in the past where students have sat for OWLs without having ever set foot inside a formal lesson.

The NEWT examination committee is under the Division of Magical Education. Perhaps your brother could pick up a pamphlet or two for you; I believe he works just down the corridor, in Communications.