There are two problems I see here. One, the Windows ACL system has nothing to do with that of Linux (POSIX actually), they are different. That said, there are things that can be done to more or less represent the same permissions in either system.
Second problem is that you don't even seem to share the same user database or ID mapping between the systems, so how do you expect to maintain any kind of permission info when you transfer files between them?
My advice is that if the ACLs aren't too complex, just replicate them by hand using setfacl. If not, you are in to an adventure into joining a windows machine and a linux machine to the same authentication domain and share the user database. If that's your wish, you can begin reading here