Better Scroll Rect
The Better Scroll Rect provides two extra fields: Horizontal Start Position and Vertical Start Position.
The Scrollbar will jump to that (relative) position on start.
If you have horizontal / vertical Scrollbars assigned, you can inject the current value of them by a click on the button below the respective start position slider.
In the background a Coroutine is started which waits a frame before changing the positions. Therefore, it is not possible in this version of Better UI to have the position change in edit mode.
Note
This implementation will probably change in a future version of Better UI by changing the pivot point of the scroll view.