In miles, feet, or kilometers, what size is Neverwinter from north wall to south wall and also from the west wall to the east side?
This PC can move diagonally. Doing so does not involve moving through a hostile creature's square.
(See PHB p.192: Variant: Playing on a Grid)
Characters (and monsters) can move diagonally like they do horizontally and vertically, as a one-square move. From the Entering a Square section we can tell that moving into a diagonal square doesn't involve moving into either of the adjacent squares that border it: with only 5' of movement left you can enter the diagonal square.
Diagonal movements can be restricted by an obstruction that completely fills the adjacent-not-diagonal squares—see the Corners section. But that section specifically calls out terrain and trees—stationary objects, in other words—as the obstruction. A medium-sized creature does not create this type of obstruction: as described on the previous page (Space), the opponent doesn't fill the square it's in.
There is no recognition of the longer path between two diagonal centers than between two truly adjacent centers. The PC can move to any of the corner-spots as one 5' portion of their movement.
The DMG has alternate rules for counting diagonals as alternately 5' and 10' (p. 252), but this only matters for distance considerations—even under that "Optional Rule: Diagonals" your PC could still move to a diagonal space.
Beware opportunity attacks, though!
Depending on how much of a back story you want:
Beggar's Nest was the poor district:
The Beggar's Nest is a district of destitution where most of the poor inhabitants live and work. To the north is a great graveyard, and there has been several records of undead attacks in this district. Under this graveyard there are unexplored tombs and crypts from deceased men. Packed with narrow streets and slum houses, this district is the most exploited of the five.
And The Dock's was the seat of most crime:
The Docks is the district most prone to criminality. It is ruled mostly by organized criminal leaders and thugs, often leading to black auctions and markets. With the main harbour to the west, many illegal wares are easily smuggled into the district, much because authorities do not have a very strong grip on the contemporal events. The district founds The Golden Apple, an inn, and Twenty in a Quiver, a local warbrand shop.
The City's Destruction:
However; much of the city was destroyed by the spellplague and the eruption of Mount Houtenow and has been rebuilt. Only three districts are available: Blacklake, which has been rebuilt in the area near Castle Never, the Docks, which remain largely unchanged, and the Merchant Quarter, which may be the remnant of the City Core.
I believe much of the poor have moved into the Southeastern quarter of the city core, or the ruined center of Blacklake District.It is conceivable that the characters have been recruited into any faction or guild in the city, or none at all.