> What are thing to take into consideration when the handover starts, to make sure we have everything and be capable to maintain that product in the future?

The things you should make sure are :

 - the code builds successfully
 - the unit tests build and all pass
 - any other tests pass (acceptance, integration, etc)
 - you get the database of open issues (easy to get if they use bugzilla, or similar)
 - the product runs (installation instructions).

Everything else is up to the current maintainer to hand over.