I must admit that I've never heard of a "sunday run" in American English (southern, north-western, or southwestern dialects). I'll take a stab at explaining how I would interpret it, but it is a novel interpretation and might not be the same as the original speaker/writer intended.
Sunday used to be a rather official "day of rest" (derived from "keeping the sabbath holy" from Judo-Christian tradition), and so to describe something with Sunday implied that it was leisurely and unhurried, slow. Thus the term "Sunday driver", which now means someone who is driving slow and leisurely as though to enjoy the scenery and is not purpose-driven on arriving somewhere quickly.
As a native English speaker, I thus assume anything that uses Sunday as an adjective must imply something leisurely, slow, and perhaps drawn-out or lengthy. The only exception is when Sunday is used to imply something more explicitly religious or formal, like "the Sunday masses" or "Sunday best" (clothes nice enough to wear to church). David Hall (no relation, I think...) pointed out "Sunday painter", which implies an amateur or nonprofessional, and while listed in the dictionary it was the first time I'd heard of that usage.
In this context of a "Sunday run", one can seem to rule out formality (unless you have to run in a suit and tie?), an amataeur run makes little sense, and I don't see how they could mean a religious run, so it seems like this is the only sensible definition left.
I would then logically conclude that the speaker means that the City Library is at least an hour or more away at a leisurely jogging/running pace; the kind of run that would be fine for leisure, as on a Sunday when you have nothing better to do, but if you don't have that much time you'd best take a taxi.
It's a highly unusual use, but I think its not 'wrong' - just perhaps novel.