You can calculate the length of the line in miles by opening the attribute table and adding a field "MILES" and then right click the miles field and click "calculate geometry" select "miles" as units and then it will populate the length for you. Then you can create a route using "length" as your measure source. (I don't think you will need a from and to field but if you get an error, just add a field "FROMMILES" or something like that and use the field calculator to populate it with 0's (your TO field is your "MILES" field))
There might be a tool that can do this for you but manually you can use the "identify route location" on the Linear Referencing toolbar to get the milepost value at each point and use calculate field (or a model builder tool with input parameters) to add it to your milepost field in your point layer.
NOTE: If your river layer contains drastic changed in elevation you will need to find the geodesic distance, I do this by overlaying my lines on a DEM but there might be an easier way.