Το σιρόπι με ανακουφίζει όταν έχω βήχα.

Το is the neuter singular definite article, meaning the.
So το σιρόπι = the syrup.

In Greek, the definite article is used very often, sometimes more often than in English. You can say just σιρόπι in some contexts, but το σιρόπι sounds natural when referring to a specific syrup or to syrup in a general, familiar sense.

Σιρόπι is neuter. You can tell from the article το.

So:

• το σιρόπι = nominative / accusative singular
• plural: τα σιρόπια

Because it is neuter singular, the article is το.

Here με means me.

It is the weak object pronoun in Greek, and in this sentence it is the direct object of the verb:

• Το σιρόπι με ανακουφίζει
  = The syrup relieves me

Greek usually places this weak pronoun before the verb, unlike English:

• Greek: με ανακουφίζει
• English: relieves me

That is the normal position for weak object pronouns in Modern Greek.

So Greek says:

• με βλέπει = he/she sees me
• με βοηθάει = he/she helps me
• με ανακουφίζει = it relieves me

This often feels unusual to English speakers, because English puts the object after the verb, but in Greek these short pronouns usually come before it.

Ανακουφίζει is the 3rd person singular present tense of ανακουφίζω, meaning to relieve / soothe / ease.

So:

• ανακουφίζω = I relieve
• ανακουφίζεις = you relieve
• ανακουφίζει = he/she/it relieves

In this sentence, the subject is το σιρόπι, which is singular, so the verb is also singular:

• Το σιρόπι με ανακουφίζει
  = The syrup relieves me

Because the subject is το σιρόπι, which is singular.

In Greek, verbs agree with their subject in person and number. Since το σιρόπι is it / the syrup, Greek uses the 3rd person singular form:

• ανακουφίζει = it relieves

If the subject were plural, the verb would change:

• Τα σιρόπια με ανακουφίζουν
  = The syrups relieve me

Όταν means when.

It introduces a time clause:

• όταν έχω βήχα = when I have a cough

It is followed here by a normal present-tense verb:

• έχω = I have

So the whole sentence structure is:

• main clause: Το σιρόπι με ανακουφίζει
• time clause: όταν έχω βήχα

Because βήχα is the accusative singular form, and after έχω (I have) Greek uses the direct object in the accusative.

The dictionary form is:

• ο βήχας = the cough

But as an object:

• έχω βήχα = I have a cough

This is a very common pattern:

• ο πυρετός → έχω πυρετό
• ο πονοκέφαλος → έχω πονοκέφαλο

So βήχας is nominative, while βήχα is accusative.

Because Greek often omits the article in expressions like have a cough, have fever, have a headache, where English may use a.

So:

• έχω βήχα = I have a cough
• έχω πυρετό = I have a fever
• έχω πονοκέφαλο = I have a headache

Using an article is sometimes possible in other contexts, but in this kind of medical/general expression, no article is very natural.

Yes, pretty much.

Greek often does not use an indefinite article in cases where English uses a/an. So literally it is closer to:

• when I have cough

But the natural English translation is:

• when I have a cough

This is a good example of how Greek and English package the same idea slightly differently.

Yes. Ανακουφίζω has the idea of relieving, easing, soothing, or making someone feel better, depending on context.

So this sentence could be understood as:

• The syrup relieves me when I have a cough
• The syrup soothes me when I have a cough
• The syrup eases my symptoms when I have a cough

The exact English wording can vary, but the Greek idea is that the syrup brings relief.

A helpful approximate pronunciation is:

To si-RO-pee me a-na-koo-FI-zee O-tan E-ho VI-ha

A few notes:

• το sounds like to
• σιρόπι has stress on ρό: si-RO-pee
• ανακουφίζει has stress on φί: a-na-koo-FI-zee
• όταν has stress on the first syllable: O-tan
• έχω is roughly E-ho
• βήχα is roughly VI-ha, where χ is the Greek sound heard in Scottish loch or German Bach

Yes, but it means something slightly different.

• Το σιρόπι με ανακουφίζει όταν έχω βήχα
  = The syrup relieves me when I have a cough

This focuses on me as the one receiving relief.

• Το σιρόπι ανακουφίζει τον βήχα
  = The syrup relieves the cough

This focuses more directly on the cough as the thing being relieved.

Both are grammatical, but they are not exactly the same in emphasis.

The structure is:

Subject + object pronoun + verb + time clause

So:

• Το σιρόπι = subject
• με = object pronoun
• ανακουφίζει = verb
• όταν έχω βήχα = time clause

Word-for-word, it looks like:

• The syrup me relieves when I have cough

Natural English:

• The syrup relieves me when I have a cough

This is a useful Greek pattern to get used to, especially the placement of short object pronouns before the verb.