My doubt is can you move to any farm/tundra boarding that particular river?
You might be overthinking it. You can only move one hex per move 'point' (but you can get (1), (2) or (3) move points). River-walk is literally just the ability to cross over the river to an adjacent hex - it doesn't allow you to 'sail' down the river to another hex somewhere else
So yes, it has to be a hex directly across the river