Assuming that you tagged the object, its tag will be in tag_ptr. Then you can use GetObjectsWithTags to get the actual jobject with that tag, and finally use the JNI call GetObjectClass to get the objects class as a jclass. Finally, use jvmti->GetClassSignature to get the actual class name as a string.
If you tagged the object's class, class_tag will have it. You can probably use this to skip a step, and get the jclass directly with GetObjectsWithTag.