Year by year when there is heavy rain in the mountains far away south, the River Nile rises and floods the fields.
Does the sentence "in the mountains far away south" mean the location of the mountains is on the south?
Yes, the mountains are far away in the south, but also this is where the rainfall occurs that ultimately floods the region by causing the river to swell. Rivers often start in a mountainous area where there is rainfall, but then carry that water to far away places.
There are a couple of things wrong, I feel, with the sentence.
The expression "year by year" means that something is continually progressing on an annual basis. For example, "she grew taller year by year". Similarly, we might also say something changes "day by day", or "little by little". As your example is talking about flooding that occurs every year, but then the waters recede only for it to happen again, this is not the appropriate expression. It should be "year after year", meaning that it repeats.
I also think the sentence may be missing another comma to make a parenthetical clause.
I would write it as:
Year after year, when there is heavy rain in the mountains far away south, the River Nile rises and floods the fields.
A parenthetical statement can be enclosed in either parenthesis (brackets, like this) or using commas. What is contained with the parenthesis is useful, but not essential information.
You could understand the example, as I have written it as:
Year after year the River Nile rises and floods the fields.
The parenthetical statement "when there is heavy rain in the mountains far away south" adds detail by telling us what precise conditions cause this to happen on an annual basis.