How soon after filing my application can I start using my patent?

Straight away. 
Keep in mind, however, that by using (‘exploiting’) the patent, you will be revealing it to the general public. 
Given this, and the fact that in Monaco it takes at least six months for a patent to appear in the Official Bulletin of Industrial Property (BOPI), it may be advisable to keep your invention secret during that time. This will help ensure you preserve your technological advantage over rivals, and are able to sign non-disclosure agreements preventing anyone from discussing your invention with third parties.