Múlt hét óta van bazsalikomunk.

Because Hungarian normally uses the present tense with óta for something that started in the past and is still true now.

So:

• Múlt hét óta van bazsalikomunk.
  literally: Since last week, we have basil
• natural English: We’ve had basil since last week

This is one of the places where English prefers the present perfect, but Hungarian usually does not.

Óta means since when it refers to a starting point in time and something continues up to the present.

So:

• múlt hét óta = since last week
• tegnap óta = since yesterday
• három óra óta = since three o’clock

It is a postposition, so it comes after the noun or time expression, not before it.

Because múlt héten and múlt hét óta are two different structures.

• múlt héten = last week as a time when something happened
• múlt hét óta = since last week

With óta, the time expression appears in the form used before that postposition, so you say:

• múlt hét óta
• not múlt héten óta

Compare:

• Múlt héten vettünk bazsalikomot. = We bought basil last week.
• Múlt hét óta van bazsalikomunk. = We’ve had basil since last week.

Bazsalikomunk is made up of:

• bazsalikom = basil
• -unk = our

So literally it means our basil.

But in Hungarian, possession is often expressed with a structure that literally looks like there is our X, while in English we translate it as we have X.

So:

• Van bazsalikomunk.
  literally: There is our basil
• natural English: We have basil

This is a very normal Hungarian way to say have.

Because the ending -unk already tells you who the possessor is: we / our.

So bazsalikomunk already contains the idea of our basil, and the full sentence already tells us we have basil.

You only add a separate pronoun or dative form if you want emphasis or contrast, for example:

• Nekünk van bazsalikomunk. = We have basil. / As for us, we have basil.

But in a neutral sentence, it is not necessary.

Because this sentence is talking about basil in an indefinite / nonspecific way, more like we have some basil.

Also, bazsalikom is often treated like a mass noun, so no article is needed here.

Compare:

• Van bazsalikomunk. = We have basil / some basil.
• Van egy bazsalikomnövényünk. = We have a basil plant.

Using a definite article here would usually make the noun sound much more specific, and that is not the normal neutral way to express this idea.

Yes, the verb to be is often omitted in the 3rd person singular present with simple noun or adjective predicates, but possession is different.

To say have, Hungarian commonly uses van plus a possessed noun:

• Van autóm. = I have a car.
• Van testvéred? = Do you have a sibling?
• Van bazsalikomunk. = We have basil.

So here, van is necessary.

Not with exactly the same meaning.

• óta means since, with the sense from that time until now
• -tól/-től means from, but it does not by itself emphasize continuation up to the present in the same way

So:

• Múlt hét óta van bazsalikomunk. = We have had basil since last week.
• Múlt héttől would sound more like starting from last week, often used for schedules, rules, or changes

For this sentence, óta is the natural choice.

Yes, Hungarian word order is flexible, but it reflects topic and focus.

Here:

• Múlt hét óta sets the time frame: since last week
• van bazsalikomunk gives the main information: we have basil

This is a natural, neutral word order.

If you change the order, the emphasis can change. For example, putting bazsalikomunk in a more prominent position can suggest contrast, like it’s basil that we have.

So the original sentence is a normal, unmarked way to say it.

Because múlt hét is the normal everyday way to say last week in expressions like this.

• múlt hét = last week
• múlt hét óta = since last week

Elmúlt can also refer to something that has passed, but in ordinary calendar expressions, múlt hét, múlt hónap, múlt év are the standard forms.

So Múlt hét óta is the most natural choice here.