Yes, Italian and English differ in this kind of conditional phrase: Italian need the futur tense in both verb-phrases, but English use present tense for the condition (When I arrive) - even though it is in the future, logically speaking. We have the same in Swedish, we look at 'When I arrive' - when all this will happen in the near future - as the 'present' when something else 'will happen'.
The actual correct answer is in the principal clause "i SHALL write to you. I know we seldom say this. Will is a secondary use meaning "with determination". I realize we do not say "shall"much in conversation, neverthelesss it is the more correct form for first person singular and plural form of the futre tense.
There are two points here; one is the main point to illustrate the difference between English and Italian in this type of sentence, ie the use of the future in both parts of the sentence in Italian and not in English. Quando arriveremo, prenderemo una tazza di caffè. = when we (will) arrive, we will have a cup of coffee. The other point is the use of "you" and not "to you" for the indirect object....this is just American English were "to" is left out in colloquial speech. Although it sounds ungrammatical to most English speakers it is a perfectly fine alternative.