Όταν προσγειωνόμαστε σε ξένη πόλη, κάθε ταξιδιώτης και κάθε ταξιδιώτισσα ψάχνουν κάτι να φάνε πριν πάνε στο ξενοδοχείο.

Here προσγειωνόμαστε is a generic we. It does not necessarily mean we, the specific speakers. It means something like:

• when we land = when people land
• when one lands
• when travelers land

Greek, like English, often uses a general we in statements about typical human behavior. Then the sentence becomes more explicit by naming the group: κάθε ταξιδιώτης και κάθε ταξιδιώτισσα.

So the shift is natural: it moves from a general scene to a specific description of travelers.

Όταν means when.

The verb προσγειωνόμαστε is in the present/imperfective because the sentence describes a general, repeated situation:

• Whenever we land in a foreign city...

Greek often uses the present in this kind of habitual statement.

If you wanted to talk about one specific future occasion, Greek would more likely use an aorist-subjunctive form, for example:

• Όταν προσγειωθούμε... = When we land... on that one future occasion

So in your sentence, the present shows something that usually happens.

Because προσγειώνομαι is one of many Greek verbs that have middle/passive endings but an active meaning.

So:

• προσγειώνω = I land something / I bring something down
• προσγειώνομαι = I land

Here:

• προσγειωνόμαστε = we land

So even though the ending looks passive to an English-speaking learner, the meaning is active.

σε can mean in, at, or to, depending on context.

Here σε ξένη πόλη means:

• in a foreign city
• or more naturally in English, in a city that is foreign/unfamiliar to us

As for the missing article: Greek often leaves out the indefinite article in expressions like this. So:

• σε ξένη πόλη = in a foreign city
• σε μια ξένη πόλη = also possible, with a slightly more explicit a/some

Both are possible, but the version without μια is very natural.

This is an inclusive masculine + feminine pair.

• ταξιδιώτης = male traveler
• ταξιδιώτισσα = female traveler

By saying both, the sentence explicitly includes everyone. This is common in Greek when the speaker wants to mention both genders clearly.

Also, κάθε is repeated because each noun has its own determiner:

• κάθε ταξιδιώτης και κάθε ταξιδιώτισσα

That is more natural than trying to make one κάθε cover both nouns.

Because κάθε is followed by the singular in Greek.

This is one of the basic rules:

• κάθε μέρα = every day
• κάθε άνθρωπος = every person
• κάθε ταξιδιώτης = every traveler

So even though the meaning is plural in a broad sense, the noun after κάθε stays singular.

Because the whole subject is not just one κάθε phrase. It is a compound subject:

• κάθε ταξιδιώτης και κάθε ταξιδιώτισσα

These two singular units are joined by και (and), so together they refer to multiple people. That is why the verb is plural:

• ψάχνουν = they look for

Compare:

• Κάθε ταξιδιώτης ψάχνει... = Every traveler looks...
• Κάθε ταξιδιώτης και κάθε ταξιδιώτισσα ψάχνουν... = Every male traveler and every female traveler look...

So the plural verb is expected.

It means something to eat.

Literally, it is closer to:

• something that they may eat
• something for them to eat

This structure is very common in Greek because Modern Greek does not use an English-style infinitive the way English does. So where English says:

• something to eat

Greek often says:

• κάτι να φάνε

The να + verb structure does a lot of the work that the infinitive does in English.

Because τρώω is an irregular verb.

Its forms come from different stems:

• present: τρώω = I eat
• aorist: έφαγα = I ate
• subjunctive based on the aorist: να φάω, να φας, να φάει, να φάμε, να φάτε, να φάνε

So here:

• να φάνε = for them to eat / that they eat

This is completely normal, even though it looks very different from τρώω.

πριν means before, and it can be followed by a subjunctive-type verb form.

In modern Greek, both of these can occur:

• πριν πάνε
• πριν να πάνε

So the version in your sentence is perfectly normal.

Here it means:

• before they go to the hotel

After πριν, Greek often uses this kind of verb form to refer to an action that has not happened yet at that point.

στο is the contraction of:

• σε + το = στο

So:

• στο ξενοδοχείο = to the hotel / at the hotel

In this sentence, because the verb is πάνε (they go), it means:

• to the hotel

This kind of contraction is extremely common in Greek:

• στο = σε το
• στη = σε τη
• στην = σε την
• στους = σε τους

This is an aspect question.

In this sentence:

• να φάνε presents eating as one complete event
• πριν πάνε presents going to the hotel as one complete event

That is why Greek prefers these aorist-type forms here.

If you used more imperfective/present-like forms such as:

• να τρώνε
• πριν πηγαίνουν

the meaning would sound more like an ongoing or repeated action, which does not fit as well here.

So the contrast is:

• προσγειωνόμαστε = general repeated situation, whenever we land
• να φάνε, πάνε = single complete actions within that situation, eat, go

This present-vs-aorist contrast is one of the most important features of Greek grammar.