This sounds like a typical description of what one might expect to see in an Oxbridge college, either in Conan Doyle's time or today (they haven't changed much). Note that "one of our great university towns" in the story must mean Oxford or Cambridge.
The "long, low, latticed window" would be something like this (latticed), only a different shape (long and low meaning small in height and large in width).
The "ancient, lichen-tinted court" would be something like this (Oxford) or this (Cambridge): a courtyard, or quadrangle (Oxbridge people call them "quads" or "courts"), basically a square of grass surrounded by paths and ancient buildings, which one might well expect to be covered in lichen. Many Oxbridge colleges have them, and rooms overlooking them in the college, as you can see in the pictures.
The "Gothic arched door" would be something like this, leading out of the professor's college room onto a worn stone staircase. Again, this is typical Oxbridge fare even today. As a college tutor and lecturer, Hilton Soames would have a set of rooms in the college, probably in an old stone building some centuries old. He has a view from his window onto the ancient quadrangle, and even the door out of his rooms onto the staircase is something old and grand, set in a Gothic arch.
See this, another typical picture from an Oxbridge college, for the kind of door and staircase we might be looking at. Note that the organisation of rooms in these colleges is generally based around staircases, which usually have only one or two doors on each storey: a typical college "address" might be Room 3 on Staircase F, or F3 for short, and that might be already on the second floor. So it also makes sense that Hilton Soames's students might have individual rooms above him on the same staircase.