The map is 28 tiles across, traveling half way across would take 2 weeks (assuming decent roads and a speed of 25). So going out and back from the center is a whole month (without doing anything else). It seems like all this travel wouldn't leave a lot time for running a kingdom. Is it expected by the time adventurers are traveling these distances, they'll have other means of transport? Is it expected characters will have access to teleport (which is uncommon) by that time? Are there other key transport spells?
The rules simply require spending a week doing the downtime activity of running the Kingdom to avoid the vacancy penalty, with no actual restrictions on where the character is when they take the activity, so it's not like they're required to spend that time in their capital (or technically even in their Kingdom) so it's reasonable to say they could be spending their time running the Kingdom from a base camp or settlement nearer the border rather than needing to spend a week each way trekking to and from wherever their current exploration target is. The vacancy penalty also isn't too terrible since it's either take the penalty listed for the role or the PC loses one of their (three) leadership activities.
PCs are also pretty much just assumed to have horses or some other form of transport faster than walking (especially since their first story-advancing encounter in the Stolen Lands has an extremely high chance of netting them four horses if they didn't already have them) and in a worst-case scenario if movement times are really becoming an issue you as the GM can either give them a faster method of movement or adjust time requirements to suit how fast your group moves so they get there at the appropriate time.
Is there a guide to which rivers are navigable? For that matter, is there a map of which river is which? They don't seem to be labeled and looking through tile descriptions for the names is tedious.
If it's shown on the map, assume it's a navigable waterway unless a specific hex description notes otherwise (at least one hex has a section of falls which are noted to not be navigable). For labels I'd suggest finding a copy of the hex map from 1e - it's the same map, but things on it are actually labeled.