Mi promesas, ke mi presos la finan dosieron morgaŭ kaj petos la estron subskribi ĝin.

Ke introduces a content clause: what is being promised.

So:

• Mi promesas = I promise
• ke ... = that ...

In English, that is often optional, but in Esperanto ke is normally kept in this kind of sentence.

Because the act of promising is happening now.

The speaker is making the promise at the present moment, so Esperanto uses -as:

• promesas = am promising / promise

What is promised will happen later, so the later actions use -os:

• presos = will print
• petos = will ask

The ending -os marks the future tense.

Both actions are future actions inside the ke clause:

• mi presos la finan dosieron morgaŭ
• kaj petos la estron subskribi ĝin

So the speaker is promising that later they will do both things.

Because the same subject, mi, applies to both verbs.

So mi presos ... kaj petos ... means:

• I will print ...
• and I will ask ...

Esperanto often leaves out a repeated subject when it is clearly the same.

Because dosieron is the direct object, and in Esperanto direct objects take -n.

Also, adjectives agree with the nouns they describe in both number and case, so fina also becomes finan.

Breakdown:

• dosiero = file
• dosieron = file as direct object
• fina = final
• finan = final, matching the object form of dosieron

So la finan dosieron is one unit: the final file.

La is the definite article, like English the.

It shows that this is a specific file, not just any file. The sentence is talking about a particular final file, so la is natural here.

Because in this structure, la estron is the direct object of petos.

Esperanto commonly uses this pattern:

• peti iun fari ion = to ask someone to do something

So here:

• petos la estron = will ask the boss
• subskribi ĝin = to sign it

Even though the boss is a person, it is still the direct object of peti, so it gets -n.

Because after peti, Esperanto normally uses the infinitive to show what someone is being asked to do.

So:

• petos la estron subskribi ĝin = will ask the boss to sign it

If you used subskribos, that would mean will sign, which does not fit after petos here.

If you used subskribu, that would be an imperative or jussive idea, more like a direct command: let him/her sign or sign! That is not the structure used in this sentence.

Ĝin refers back to la finan dosieron.

The base pronoun is ĝi, used for a thing. Since it is the object of subskribi, it takes -n:

• ĝi = it
• ĝin = it as direct object

So subskribi ĝin means to sign it, with it = the final file.

In this sentence, the most natural reading is that morgaŭ applies to the whole future plan, so probably both actions happen tomorrow.

That is:

• tomorrow I will print the final file
• and tomorrow I will ask the boss to sign it

If a speaker wanted to make the timing different or extra-clear, they could reword the sentence.

Yes. Esperanto word order is fairly flexible because the endings show the grammatical roles.

For example, Morgaŭ mi presos la finan dosieron kaj petos la estron subskribi ĝin would also be correct.

The given order is simply a natural, neutral way to say it.

A comma before a ke-clause is very common in Esperanto.

It helps mark the start of the subordinate clause:

• Mi promesas, ke ...

You may sometimes see punctuation used a little differently by different writers, but this comma is standard and very natural.