Minä palaudun nopeasti viikonlopun jälkeen.

Yes, it can be left out. Finnish often drops subject pronouns because the verb ending already shows the person.

So palaudun already means I recover / I bounce back.
That means Palaudun nopeasti viikonlopun jälkeen is a perfectly normal sentence.

Adding minä makes the subject extra clear or slightly emphatic: I recover quickly after the weekend.

Palaudun is the 1st person singular present tense form of palautua.

In this kind of sentence, palautua means something like:

• to recover
• to bounce back
• to return to normal
• to get your energy back

So the idea is not just physical movement back somewhere. It is more about recovering your condition, energy, or balance.

This is a normal Finnish verb change.

The dictionary form is palautua. In the present tense, the stem changes to palaudu-, and then the personal ending -n is added for I:

• palautua = to recover
• palaudu- = present-tense stem
• palaudun = I recover

So this is not random; it is part of the verb’s conjugation pattern.

It is in the present tense.

In English, the most natural meaning is probably a habitual or general one:
I recover quickly after the weekend.

Finnish present tense can also sometimes refer to the near future depending on context, but here it most naturally sounds like a general fact about the speaker.

Because nopeasti is an adverb, while nopea is an adjective.

• nopea = fast, quick
• nopeasti = quickly

Here the word describes how the recovering happens, so Finnish uses the adverb form: nopeasti.

A very common pattern is:

• adjective + -sti → adverb

For example:

• hidas → hitaasti = slowly
• varma → varmasti = certainly / surely

This is a very common Finnish structure.

• viikonlopun = of the weekend / weekend in the genitive form
• jälkeen = after

Together, viikonlopun jälkeen means after the weekend.

The important thing to notice is that jälkeen is a postposition, not a preposition.
That means it comes after the noun phrase, not before it.

English: after the weekend
Finnish: the weekend + after → viikonlopun jälkeen

Because jälkeen requires the genitive case.

The basic form is:

• viikonloppu = weekend

But before jälkeen, it becomes:

• viikonlopun = weekend in the genitive

So:

• viikonlopun jälkeen = after the weekend
• tunnin jälkeen = after an hour
• ruoan jälkeen = after the food / after eating

This is something you often just have to learn together with the postposition: jälkeen + genitive.

It literally means week-end.

It is made of:

• viikko = week
• loppu = end

So viikonloppu is literally the end of the week, just like English weekend.

In the sentence, the genitive form is viikonlopun.

No, Finnish word order is fairly flexible.

The version here is a neutral, natural order:

Minä palaudun nopeasti viikonlopun jälkeen.

But you could move things around for emphasis, for example:

• Viikonlopun jälkeen palaudun nopeasti.
• Nopeasti palaudun viikonlopun jälkeen.
  This is possible, but less neutral.

Even when the order changes, the cases and verb endings help show what the sentence means.

Finnish does not have articles like a or the.

So viikonloppu or viikonlopun can mean:

• a weekend
• the weekend
• sometimes just weekend

The exact meaning depends on context.

In this sentence, English usually wants the weekend, but Finnish does not need a separate word for that.

Yes, palaudun is natural if you mean I recover my energy or I bounce back after something tiring.

It is especially suitable for things like:

• a busy weekend
• hard work
• exercise
• stress

So the sentence suggests that the weekend takes something out of the speaker, and then they recover quickly afterward.

Depending on context, other verbs are also possible, but palaudun is a good and natural choice for this meaning.