As mentioned by @Jacob F in the comments. A good starting point would be the Euclidean Distance Tool.
Basically what you would need to do is use the Rivers as the input dataset. This will then create a raster output where each cell (pixel) represents the distance it is from the river / rail or whatever feature you have decided to use. This distance is to the closest part of the input layer.
To get this into a table, I would suggest converting your clipped raster (your study area) into a point shapefile. The Raster to Point Tool, located in the Conversion Toolbox will accomplish this. Taken directly from help:
For each cell of the input raster dataset, a point will be created in
the output feature class. The points will be positioned at the centers
of cells that they represent. The NoData cells will not be transformed
You can then use the "Extract Values to Points" Tool located in the Extraction Toolbox of Spatial Analyst, to extract the "Distance" values from the Euclidean Raster created from the Rivers. Your output will be a new Shapefile containing the points from your study area raster and an attribute table with the distance each point (cell) is from the river. You can then export this attribute table to *.dbf / *.txt for consumption in Excel or another spreadsheet type software package.