I have a client with a self-hosted 2010 Exchange Server. They want to have Auto Reply automatically turn on only between 5:00 PM and 8:00 AM. During these hours they do not want any notifications of any new emails to appear on their devices, but would like to still be able to check their email.
Is exchange configurable to do any of this or is there some software that can do this out of the box? Or will I need to have a custom solution developed?
Out of the box no, but you can easily code that with EWS.
An example in C# to Set Out of Office message for one mailbox;
With that example, you can see it's quite simple, you just have to use an admin's account that can impersonation all the mailbox to set the out of office message.
If you don't code, you can send with curl some XML too. Like shown there: Quick and Dirty UNIX Shell Scripting with EWS
Example from there
For the device notification on the other hand is something else, it's the device itself that do the push notification, not Exchange. For that part I can't help.