How far upstream would severe sea level rise (1m) noticably affect a large river?
This is both a simple and complex answer.

The simple bit is that a 1m rise in sea level would impact the river to a point 1m higher than the current point (with a slight error margin dependent on flow rates etc)

The rise itself over the next 85 years will not push back along the river (as Michael mentioned) as it's not like a tidal bore such as the Severn Bore. 

The complex bit is that the 1m rise could be along an estuary, or across winding wetlands, or a wide delta or other feature, so the change to river flow could actually be quite catastrophic. And these effects could impact very far upstream, especially in the case of a river such as the Rhine, with a heavily populated floodplain. Wikipedia notes the last 50km of the Rhine are at 0m elevation, so that's an incredibly large possible impact area.