Is it possible to have a root-running app on no-root android? I have an app which needs su privileges, But my device should not be working on rooted. I want to know if there is any way to take the access to just one app. For instance root the device, set the privileges for that app, unroot the device. Is it possible?
Firstly let me clarify something.

Being rooted is not a binary situation where you are either rooted or not rooted. Process of rooting is adding custom binaries to system partition. These binaries, depending on the way they are coded, can answer suid calls from apps so those apps can have the right to execute orders not with their own UID but with the UID 0 that is the root user. 

Secondly, can you run apps with su rights without adding those binaries?

It can be possible but even if it is, I don't know of such a method. Maybe there are workarounds but I never heard of them. So I think I can say no to that.