There are ways to implement routing although it requires a custom mobile app and the use of some other ESRI technologies. There is no way built-in to the Mobile ADF/API itself.
I have never done either of these, but have researched them at a high level:
1. Call an ArcGIS Server service to get route and directions, and display route over Mobile map. Would require a network connection to call the server. Custom solution.
2. Use ArcGIS Engine and Network Analyst, and a local network dataset, to create a local routing component. Then call the Engine component from Mobile. Would require a GDB with networked dataset (can't be part of the mobile cache). Obviously a completely custom solution, and not trivial. And of course would require ArcGIS Engine licensing in addition to mobile, so potentially expensive if you have large deployments. And also would require a Windows device (laptop or other) and a fairly robust one at that, to run Engine.
Those are the 2 approaches I know of. #1 actually is not particularly hard if you are a developer or have access to developers, and a custom mobile app is acceptable in your situation. #2 is a significant effort and a very "heavy" solution for sure. No easy answer to this one....
