For a thourough and fast cleaning of a case, an ultrasonic bath is the best solution.
Uultrasonic bath, "budget"-type
With strong sonic waves above 30kHz, the dirt is “blown out” (cavitation). In combination with warm water and cleaning solutions, even massively dirty pieces can be cleaned without the need of physical force. This is very good for thin gold coated cases.
One thing is important: Only clean empty, open cases. But this is self-evident, since the hardest dirt is on the screwing or pressing of the back.
The costs for an ultrasonic bath are from 20 Euro up to several hundered or thousand Euros. The first ones are more like toys and are rather soft ones, but they nevertheless clean much better than no ultrasonic bath. Of course, a real watchmaker will use a professional device, which is much stronger and contains a built-in heating, but these devices are not for laymens, since they can harm your fingers and hands if you use them in a wrong way. A 20 Euro device is pretty safe.
A tip: After a warm ultrasonic bath, remaining dirt parts can easily be rubbed off with a toothbrush.