Ronald Dale Karr's book, The Rail Lines of Southern New England and Robert M. Lindsell's book, The Rail Lines of Northern New England should be your two best friends for looking this sort of thing up. They have a short history of the line in the 'Southern New England' book as well as a general map.
Another best friend you should have is the online Maptech historical USGS map collection at
http://historical.maptech.com/
Or, you can try
http://docs.unh.edu/nhtopos/nhtopos.htm to access the University of New Hampshire's topo map collection. Basically the same thing as the Maptech collection, except that they only cover the New England states and New York.
Both sites provide the same information.
Although you might not find any ROW remains in the downtown Newburyport area (the old ROW may now be an alley, street, or just built over) - - explore a bit around the Newburyport station. At one time, the Boston and Maine Newburyport Branch crossed the Eastern Railroad there -- and the branch you are looking for also left the B&M Newburyport Branch at a switch right near the station, crossed the Eastern Railroad and hooked around the east side of Newburyport, eventually running right along the wharves. The entire branch forms a sort of question mark shape around Newburyport.
I'd be willing to bet that if you take a look around at Newburyport, you might find the remains of the beginning of this branch. Good clues to look for are telephone or electrical lines (they often follow old ROWs) or sewer/water lines (identified by a 'line' of manhole covers).
Good luck.