if you want a pub that stays put while the world changes around it, look up a list of England's scheduled buildings/monuments for London (or wherever you pick to put it.) this is basically their version of a historic building register, and it includes a fair number of pubs. these are buildings that the government recognizes as protected historic sites, and prevents the destruction or extensive renovation of them. in fact when they end up damaged due to age, weather, etc.. they require the repairs to be in the same methods and materials as it was originally (or as close as modern times can get.)
some of these sites go back to medieval times. it might not be hard to find one that dates back far enough you can claim it was built to replace something even older, from before good records were kept. and using a real location (even if you end up fictionalizing the name and story) makes it easier to look up what sort of real events might have happened on that spot, which could give some story ideas.
you could even have the site be a protected on in the story, with the immortal perhaps having worked to ensure it was recognized as such so that they could work a little less hard about keeping it out of the hands of land developers wanting to tear it down.
(though those have been a major danger lately.. there have been a number of incidents where a developer bought a building, found out it was a protected building and they couldn't rip it down or rebuild it into a modern monstrosity, and then "mysterious fires" broke out along with "odd barriers preventing firefighters from reaching the building".)
