The archer wins
Assuming they actually hit the spellcaster, that is.
From "Ready" (PHB, 193):
…you can take the Ready action on your turn so that you can act later in the round using your reaction.
So, let's see what a "reaction" does (PHB, 190):
… If the reaction interrupts another creature's turn, that creature can continue its turn right after the reaction.
From these parts of the rules as written, we can see that clearly a reaction can interrupt the turn of another creature. Indeed, we see this in Opportunity Attacks where the reaction (the Opportunity Attack) interrupts the movement of the other creature, which they can then continue after the attack has been resolved.
Likewise, the reaction of the archer, their readied action triggered by the appearance of the spellcaster happens immediately and interrupts whatever else was going on, regardless of whose turn it is.
Since the trigger is not the completion of the spell, but rather the appearance of the spellcaster, and the rules clearly explain that the appearance happens the instant the casting of the new spell begins, and since spellcasting isn't instantaneous, the archer's arrow is loosed before the casting of the spell is complete.
There is clearly a wide range of opinion on this topic. It has been addressed, at least tangentially, in many different forms. See e.g. Timing of the Ready action when the trigger is associated with spell casting?, Can a caster interrupt another caster from casting Time Stop by readying a Time Stop of their own?, Is it possible to interrupt spellcasting?, and Do reactions interrupt their triggers or not?.
There is a certain level of debate even within those questions and answers. Certainly, I would say a majority feel that spell casting cannot be interrupted except by powers which specifically say that it can be (i.e. do not agree with my view). But there is no consensus. More to the point, at least one opposing view has a direct statement from Jeremy Crawford to back that view up.
Quoted one of the answers to one of those many related-if-not-duplicate posts is a tweet from him:
Can silence interrupt a spell caster? I.e. cleric readies silence, trigger is enemy begins spell casting.
As DM, I'd allow a readied silence spell to interrupt the casting of a spell with a verbal component.
I'm obviously biased, but to me this clearly means that casting a spell is something that can be interrupted by a readied action. It's not a long walk from there to determine that any readied action, especially when the trigger is "becomes visible" (which in turn happens the very moment the casting begins), rather than (for example) "casts a spell" (which I readily agree could be interpreted as the entire action of casting), is able to interrupt the casting of a spell, when the trigger is worded appropriately.