vintage servicing

[fat-sam]

i recently had my dj serviced but was unhappy with the results with regards customer service and a few new scraches that developed, it was also losing 5 mins a day, so i took it to a less expensive but more knowledgable guy as opposed to the one recomended by rolex in bond st. and he said he had done an ok job servicing it, but had neglected a few obvious thing which he would fix there and then for no charge(top bloke )- it now keep better time, although still not great(or cosc standard as it should).

i made a chance phonecall to rolex sevice centre in st james sq after the recomendation of someone on here, and i was surprised to find that they said there was a good chance they would be able to service it there, as many watches from early fifties used movements thay still worked on.
mine looks identical(aside from the semi bubble back) to a modern DJ- so before i take a day off work and head in there, is there a way to tell if mine is suitable?

sorry for the longwinded way i ask but how do you find out what model/movement number a watch has?

can you do it without opening it up?