Πριν φύγω για τη δουλειά, βάζω το μεσημεριανό μου στο τάπερ.

Πριν means before.
φύγω is the perfective subjunctive form of φεύγω (to leave).

So Πριν φύγω means before I leave.

A very important point for learners: φύγω here is not a past tense, even though it looks like an aorist-based form. After πριν, Greek often uses this kind of form to refer to a single event that is about to happen relative to the main clause.

Because Greek makes an aspect distinction:

• φεύγω = imperfective / ongoing / general leaving
• φύγω = perfective / one complete act of leaving

After πριν, when you mean before I do the one act of leaving, Greek normally uses the perfective form:

• Πριν φύγω = before I leave

Using φεύγω here would not sound natural in this sentence.

Greek often leaves out subject pronouns because the verb ending already shows the person clearly.

Here:

• φύγω = I leave
• βάζω = I put

So Greek does not need εγώ unless you want emphasis or contrast.

For example:

• Εγώ βάζω το μεσημεριανό μου στο τάπερ, όχι εκείνος.
  = I put my lunch in the container, not him.

Without emphasis, leaving εγώ out is completely normal.

Literally, it means for the work/job, but in natural English here it means for work or to go to work.

So:

• φεύγω για τη δουλειά = I leave for work

This is a very normal Greek way to express going off to work.

Greek uses the definite article much more often than English does.

So where English says:

• go to work
• come home
• at school

Greek often says:

• πάω στη δουλειά
• έρχομαι στο σπίτι
• είμαι στο σχολείο

So τη δουλειά is completely natural Greek, even though English would not say the work in this context.

Because the final -ν of την is often dropped before certain consonants. Before δ in δουλειά, τη δουλειά is the normal form.

So:

• τη δουλειά = normal here

You will often see this pattern with the feminine accusative article:

• τη φίλη
• τη μέρα
• τη δουλειά

This is something learners see very often in everyday Greek.

Because the sentence describes a routine or habitual action.

Greek uses the simple present for habits, just like English can:

• Πριν φύγω για τη δουλειά, βάζω το μεσημεριανό μου στο τάπερ.
• Before I leave for work, I put my lunch in the container.

This means something like this is what I usually do.

If you wanted a one-time future action, you would normally use:

• θα βάλω = I will put

Here το μεσημεριανό means lunch or the midday meal.

It comes from μεσημέρι (noon / midday) and can be used:

• as a noun: το μεσημεριανό = lunch
• as an adjective: μεσημεριανό φαγητό = midday meal / lunch food

In this sentence, it is a noun because it has the article το.

Because Greek usually places these short possessive forms after the noun phrase:

• το μεσημεριανό μου = my lunch
• το σπίτι μου = my house
• η τσάντα μου = my bag

So μου here means my, not to me.

This word order is very standard in Greek.

τάπερ is a very common everyday Greek word for a plastic food container, lunch container, or what English speakers might call a Tupperware container.

It comes from the brand name Tupperware, but in Greek it is often used as a general word, not just for that brand.

So:

• στο τάπερ = in the container / in the lunch box

Because στο is the contracted form of:

• σε + το = στο

This contraction is standard and extremely common in Greek:

• σε το σπίτι → στο σπίτι
• σε τη δουλειά → στη δουλειά
• σε το τάπερ → στο τάπερ

So στο τάπερ simply means in the container or into the container.

Because Greek σε covers both ideas that English often separates as in and into.

So:

• είναι στο τάπερ = it is in the container
• το βάζω στο τάπερ = I put it into the container

The verb tells you whether the meaning is location or movement.
With βάζω (put), the meaning is clearly into.

They are in the accusative, because both για and σε take the accusative in Modern Greek.

You can see that most clearly in:

• τη δουλειά → accusative feminine singular
• το τάπερ → accusative neuter singular

With neuter nouns, nominative and accusative often look the same, so the article helps you see the structure.

Because Πριν φύγω για τη δουλειά is a time clause placed before the main clause.

Greek normally separates that introductory clause with a comma:

• Πριν φύγω για τη δουλειά, βάζω το μεσημεριανό μου στο τάπερ.

This is similar to English punctuation in sentences like:

• Before I leave for work, I put my lunch in the container.

Greek word order is more flexible than English word order, although the version you have is the most neutral and natural one.

The sentence as given:

• Πριν φύγω για τη δουλειά, βάζω το μεσημεριανό μου στο τάπερ.

is a straightforward, neutral way to say it.

Other orders are possible for emphasis, for example:

• Πριν φύγω για τη δουλειά, το μεσημεριανό μου το βάζω στο τάπερ.

That version gives extra emphasis to το μεσημεριανό μου.

So the original sentence is the best basic pattern for a learner, but it is not the only possible order.

Yes, you could, but the meaning shifts slightly.

• Πριν φύγω για τη δουλειά = before I leave for work
• Πριν πάω στη δουλειά = before I go to work

The version with φύγω focuses more specifically on the moment of departure.
The version with πάω is a bit broader and less tied to the exact act of leaving.

So both can work, but the original sentence is more specifically about what happens right before leaving the house for work.