Subject and Object

Harry--subject
cries--verb
time--object

I--subject
wrote--verb
letter--object

In answer to your question, yes, "letter" is receiving the action of the verb "wrote."

These verbs are active voice, so the subject does the action, and if the verb is transitive, then the object receives the action.

Imagine if the verb in the second sentence was passive voice. The sentence would be different.

A letter was written by me to my uncle.

In this case, "letter" would be the subject of the passive voice verb "was written."

In either case, a subject of any verb would never also be the object of a preposition. For example, "by me" or "to my uncle" are prepositional phrases with "me" or "uncle" as objects of the prepositions "by" or "to."