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.
This implementation will probably change in a future version of Better UI by changing the pivot point of the scroll view.