En el sobre, el apellido de la destinataria estaba bien, pero el del remitente no.

En el sobre means on the envelope or literally in the envelope, but in this context it refers to the writing on the envelope.

It is placed first to set the scene: we are talking specifically about what appeared on the envelope. Spanish often puts this kind of location phrase at the beginning when it gives useful context.

The comma is natural here because En el sobre is an introductory phrase.

Because the recipient is female.

• el destinatario = the male recipient / addressee
• la destinataria = the female recipient / addressee

Spanish marks gender in many nouns referring to people, so the ending changes from -o to -a here.

Apellido means surname or last name.

So:

• el apellido de la destinataria = the recipient’s surname
• el apellido del remitente = the sender’s surname

A learner might confuse apellido with nombre, but they are different:

• nombre = first name / given name
• apellido = surname / family name

This is just the normal combination of de + article:

• de + la = de la
• de + el = del

So:

• de la destinataria
• del remitente = de el remitente

Spanish always contracts de el to del.

Because Spanish uses estar bien to mean to be correct, to be fine, or to be in the right state.

Here, the surname written on the envelope was correct, so:

• estaba bien = was right / was correct

You would not normally say era bien in this context.
Also, bien works adverbially here, describing the condition or correctness of what was written.

Because bien is the idiomatic choice with estar when something is correct.

• está bien = it is right / correct / fine
• es bueno = it is good

In this sentence, the idea is not that the surname was good, but that it was correctly written. So estaba bien is the natural expression.

It stands for el apellido del remitente.

Spanish often avoids repeating a noun if it is already clear. So instead of saying:

• el apellido de la destinataria estaba bien, pero el apellido del remitente no

the sentence shortens the second part to:

• ..., pero el del remitente no

Here, el means the one and refers back to apellido.

So el del remitente means the sender’s one, in natural English: the sender’s surname.

Because no replaces the repeated idea wasn’t right or was not.

The full version would be:

• ..., pero el del remitente no estaba bien

Spanish often leaves out repeated words when they are obvious from context. This is very common and natural.

So the final no means something like:

• but the sender’s wasn’t
• but the sender’s was not correct

Because remitente is usually a common-gender noun. That means the noun itself often stays the same, and the article shows the gender:

• el remitente = the male sender
• la remitente = the female sender

In this sentence, del remitente shows a masculine sender, or it may simply be the sender referred to in the masculine form.

Yes. A fuller version would be:

• En el sobre, el apellido de la destinataria estaba bien, pero el apellido del remitente no.

That is completely grammatical. The actual sentence sounds more natural because it avoids repetition.

Spanish very often uses this structure:

• el de...
• la de...
• los de...
• las de...

to mean the one(s) of... or the ... one(s).

Because the subject is el apellido, which is singular.

Even though there are two people involved — the recipient and the sender — the thing being described in the first clause is just one surname:

• el apellido de la destinataria estaba bien

So the verb must be singular: estaba.

Yes. Sobre is the normal word for envelope in Spain and in general Spanish.

So:

• en el sobre = on the envelope / in the envelope

In context, it usually refers to the information written on the outside of the envelope.

Yes, you could say:

• ...estaba correcto...

but estaba bien sounds more natural and everyday in this context.

Compare:

• estaba bien = it was right / correct
• estaba correcto = it was correct

Both are understandable, but bien is very common in ordinary speech for checking whether something is written, done, or said properly.