Translation:I need a battery for my wristwatch.
Brakhorloĝo (brak + horloĝ + o): "brako" means arm + "horloĝo" means clock + o indicates a noun.
Would a pocket watch be called something else, since it isn't an "arm clock"?
Yes, a pocket watch is called a poŝhorloĝo, just like poŝtelefono means pocket phone and is the equivalent of mobile phone. Poŝo means pocket, as we have already learnt.
So 'manhorloĝo' would br a hand watch, if you wanted to say tbat for some reason?