If I exile Rout with Narset's ability, when can it be cast?
It's a cost increase, so you may use it in combination with Narset's alternative cost.



The cost to cast a spell or activate an ability is calculated as follows[CR 601.2e]:

total cost to cast
= mana cost or alternative cost
+ additional costs and other cost increases
- cost reductions

Rout's ability adds to the total cost, so it's a cost increase. It's not an additional cost since it doesn't use the word "additional"[CR 117.8], so it's just an generic cost increase[1]. It's not an alternative cost since those replace the mana cost. Alternative costs usually include the phrase "rather than pay" or "without paying its mana cost"[CR 117.9].

So, the total cost to cast is {2}. (Alternative cost of free, plus a cost increase of {2}.)



Another example of a cost increase: Thalia, Guardian of Thraben.




To my knowledge, there's nothing that distinguishes between additional costs and other cost increases. They're the same except in name.





  601.2e [...] The total cost is the mana cost or alternative cost (as determined in rule 601.2b), plus all additional costs and cost increases, and minus all cost reductions. [...]
  
  117.8. Some spells and abilities have additional costs. An additional cost is a cost listed in a spell’s rules text, or applied to a spell or ability from another effect, that its controller must pay at the same time that player pays the spell’s mana cost or the ability’s activation cost. A cost is an additional cost only if it’s phrased using the word “additional.” [...]
  
  117.9. Some spells have alternative costs. An alternative cost is a cost listed in a spell’s text, or applied to it from another effect, that its controller may pay rather than paying the spell’s mana cost. Alternative costs are usually phrased, “You may [action] rather than pay [this object’s] mana cost,” or “You may cast [this object] without paying its mana cost.” [...]