Learn Hardcore Portuguese (Portugal): Um dia, no fundo, eu próprio vou agradecer por não ter desistido quando tudo parecia difícil. - One day, deep down, I myself will be grateful for not having given up when everything seemed difficult.

It adds an emotional or reflective nuance: it suggests that, even if on the surface you might not feel grateful yet, at a deeper, more honest level you will.

Why is eu próprio used? Does it mean I will thank myself?

Eu próprio literally means I myself. It is used for emphasis:

eu = I
próprio = own / itself / himself / herself, etc.

Here, eu próprio means:

I myself
me personally

It does not automatically mean you are thanking yourself. It just emphasizes the subject eu. In English you might say:

One day, deep down, I myself will be grateful...
One day, deep down, I personally will be thankful...

Whether you are thanking someone else, thanking life, or just feeling grateful in general is left implicit.

Note that próprio agrees in gender and number:

eu próprio (speaker is male)
eu própria (speaker is female)
nós próprios / próprias, eles próprios, elas próprias, etc.

What is the difference between vou agradecer and agradecerei?

Both are future forms of agradecer (to thank / to be grateful):

vou agradecer = I am going to thank / I will thank
agradecerei = I will thank

Vou agradecer is the periphrastic future: ir (to go) in the present + infinitive.

Agradecerei is the synthetic/simple future.

In modern European Portuguese:

vou agradecer is much more common in everyday speech.
agradecerei sounds more formal, literary, or old-fashioned in many contexts.

So the sentence could be:

Um dia, eu próprio vou agradecer... (most natural in speech)
Um dia, eu próprio agradecerei... (more literary/formal)

Why is it por não ter desistido and not por não desistir?

Both are grammatically possible, but they express slightly different nuances.

por não desistir ≈ for not giving up (more general, ongoing or habitual idea)
por não ter desistido ≈ for not having given up (a completed action in the past)

In this sentence, you are imagining a future time (Um dia, vou agradecer...) when you will look back to a past moment where giving up was an option, but you did not give up. That is a completed decision in the past.

So por não ter desistido is more precise here, because:

The gratitude in the future is for a decision that will already be in the past.

What tense or form is ter desistido, and when is it used?

Ter desistido is the infinitivo composto (compound infinitive):

ter (infinitive of to have) + desistido (past participle of desistir)

It is often used:

After a preposition (like por, de, sem)
To express an action that happened before another action or point in time.

In English it usually corresponds to something like:

for having given up
after having given up
without having given up, etc., depending on the preposition.

In this sentence:

Future action: vou agradecer (I will be grateful)
Earlier action: não ter desistido (not having given up)

So the form ter desistido clearly places the not giving up as a prior, completed event relative to the future gratitude.

Who is the subject of não ter desistido? Why is it not written?

The understood subject of não ter desistido is the same as the main subject, that is eu.

Longer, more explicit versions would be:

Um dia, eu próprio vou agradecer por eu não ter desistido...
Um dia, eu próprio vou agradecer por não ter desistido eu... (possible, but less natural-sounding)

In Portuguese, when the subject of an infinitive clause is the same as the main subject, it is very common to omit it if the meaning is clear.

If the subject were different, you would normally make that visible. For example:

Vou agradecer por tu não teres desistido.
I will be grateful that you did not give up.

Here, the subject of não teres desistido is tu, clearly marked by teres.

What is the difference between por não ter desistido and por não teres desistido?

Both are related to ter desistido, but:

por não ter desistido
– subject is understood from context, usually the same as the main clause (eu here).
por não teres desistido
– this uses the infinitivo pessoal (personal infinitive), so the verb shows who the subject is:
teres tells you the subject is tu.

Compare:

Vou agradecer por não ter desistido.
I will be grateful for not having given up. (subject = I)
Vou agradecer por não teres desistido.
I will be grateful that you did not give up. (subject of teres desistido = you, singular informal)

You can do similar things with other persons:

por não termos desistido (we)
por não terem desistido (they / you-plural)

Why is it quando tudo parecia difícil and not a different past tense?

Parecia is the pretérito imperfeito (imperfect tense). It is often used to:

describe a background situation, a state, or something ongoing in the past
express how things used to be, not a single, completed event

So quando tudo parecia difícil suggests:

when everything seemed difficult,
when everything was feeling difficult (for some period of time)

Using the perfect tense (quando tudo pareceu difícil) would sound odd here, because that would suggest a short, completed event when things suddenly seemed difficult once and were done. The imperfect fits better with the idea of a challenging period or phase.

Why are there commas around no fundo?

The commas show that no fundo is a parenthetical adverbial expression – an extra comment inserted into the sentence, not essential to its grammar.

Um dia, no fundo, eu próprio vou agradecer...

This is similar to English:

One day, deep down, I myself will be grateful...

Without changing the meaning, you could also say:

Um dia, eu próprio, no fundo, vou agradecer...
No fundo, um dia, eu próprio vou agradecer...

You can move no fundo, and you would still normally separate it with commas, because it is more of a comment on the whole idea than a strict part of the core structure.

Does Um dia mean literally one day, or more like “someday”?

In this context, Um dia is idiomatic and usually means:

someday
one day (in the future)

It does not mean a specific calendar day. It expresses a vague, non-specific future time.

If you wanted to emphasize the more literal meaning (on a particular day), context and possibly more detail would usually make that clear, for example:

Um dia de verão, fomos à praia.
One summer day, we went to the beach.

How does agradecer por work here? Is por necessary?

The verb agradecer can combine with:

a (to) – introducing the person you thank
por (for) – introducing the reason or thing you are thankful for

Examples:

Agradeço a ti. – I thank you.
Agradeço por tudo. – I am thankful for everything.
Agradeço-te por tudo. – I thank you for everything.

In the sentence:

...vou agradecer por não ter desistido...

por não ter desistido is the reason for the gratitude:
I will be grateful for not having given up.

If you removed por here (vou agradecer não ter desistido), it would sound strange or unnatural. With agradecer, por is the normal way to introduce the cause/reason.

Is eu próprio specifically European Portuguese? What would be more common in Brazil?

Eu próprio is fully normal and very common in European Portuguese.

In Brazilian Portuguese, learners more often encounter:

eu mesmo (I myself)
eu próprio also exists but is less frequent in everyday speech and can sound a bit more formal or emphatic.

So:

PT‑PT (Portugal): eu próprio is very natural.
PT‑BR (Brazil): eu mesmo is often the default in casual speech, although eu próprio is still understood.

Since you are learning Portuguese from Portugal, eu próprio is exactly what you would expect to see in native usage there.