I've just been comparing the maps of the area, one from 1852 and one modern. Unless there are inaccuracies, it seems that Long Close Lane (to the south of the pub) was moved southwards.
There is only one substantial building that is between Long Close Lane and Hope Street. I guess that has to be the Brown Cow, but it would seem that the original building was bigger than the current one. Was it partly demolished, or maybe completely rebuilt?
This is a photo screen shot of the old map of the area. I couldn't find a way of nicking the original online:
Here is the url for the old map of York if you want to check for yourself:http://www.york1852.org/
Here's the modern map:https://www.google.co.uk/maps/place/The+Brown+Cowfirstname.lastname@example.org,-1.0733174,183m/data=!3m1!1e3!4m2!3m1!1s0x0:0xe6ea7ce4c3b6829
Re: the two shots below, the top one is definitely Hope Street, as was, looking west, with a right-hand bend. it now has a left-hand bend that begins by the pub. The bottom one is Long Close Lane, as was, also looking west.