У шафі є не тільки банка меду, а й банка цукру.

У шафі means in the cupboard / in the cabinet / in the closet, depending on context.

After у / в when talking about location (in, at, inside), Ukrainian uses the locative case.
So:

• шафа = cupboard / cabinet / wardrobe
• у шафі = in the cupboard/cabinet

The ending -і shows the locative here.

Here є means there is / there exists.

So:

• У шафі є банка меду = There is a jar of honey in the cupboard

This is a very common use of є to express existence.

A useful thing to know: in Ukrainian, the present tense of to be is often omitted in simple statements like Він студент (He is a student). But when you mean there is / there are, є is commonly used.

This is the Ukrainian pattern for not only ... but also ...

So:

• не тільки банка меду, а й банка цукру
  = not only a jar of honey, but also a jar of sugar

A few points:

• не тільки = not only
• а й = but also

The й is just a variant of і (and) used for smoother sound. So а й is closely related to and also / but also.

Because after a container noun like банка (jar), Ukrainian usually puts the substance in the genitive case.

So:

• банка меду = a jar of honey
• банка цукру = a jar of sugar

Here:

• мед → меду (genitive)
• цукор → цукру (genitive)

This is very similar to English of honey, of sugar, though in Ukrainian the relationship is shown mainly by the case ending rather than by a separate word like of.

Банка is in the nominative singular, because it is the main noun in each noun phrase:

• банка меду
• банка цукру

In an existential sentence with є, the thing that exists is typically in the nominative:

• У шафі є банка меду = There is a jar of honey in the cupboard

So the jars are the things being said to exist.

It is repeated because the sentence is talking about two separate jars:

• a jar of honey
• a jar of sugar

So не тільки банка меду, а й банка цукру clearly means:

• not only a jar of honey, but also a jar of sugar

If you removed the second банка, the sentence would sound more elliptical and could suggest that the second item is being understood from context. Repeating it is clear and natural.

Шафа can mean a wardrobe, cabinet, cupboard, or sometimes a larger closet-like piece of furniture, depending on context.

So У шафі could be translated in different ways:

• in the cupboard
• in the cabinet
• in the wardrobe

If the sentence is about food like honey and sugar, English would often choose cupboard or cabinet.

Ukrainian word order is flexible, but this order is very natural for introducing something that exists in a place:

• У шафі є ... = In the cupboard there is ...

This structure puts the location first, then tells you what is there.

It sounds especially natural when you are answering a question like:

• Що є у шафі? = What is in the cupboard?

Other word orders are possible, but this one is straightforward and common.

Yes, В шафі is also possible.

У and в usually mean the same thing here. Ukrainian often chooses between them for euphony—that is, for smoother pronunciation.

So both are possible:

• У шафі є ...
• В шафі є ...

Speakers choose the one that sounds better next to surrounding sounds. In this sentence, У шафі is completely natural.

They are related, but not always identical in tone.

In the pattern не тільки ..., а й ..., а й is the standard, compact way to say but also.

So this sentence naturally uses:

• не тільки ..., а й ...

You may also see similar patterns with:

• не лише ..., а й ...
• не тільки ..., але й ...

These are all close in meaning. For a learner, the safest thing is to memorize не тільки ... а й ... as a fixed pairing.

Yes. Не лише also means not only.

So you could say:

• У шафі є не лише банка меду, а й банка цукру.

This means essentially the same thing.
The difference is mostly one of style or preference:

• не тільки = very common, neutral
• не лише = also common, slightly more formal/literary in some contexts

Ukrainian has no articles like a/an/the.

Whether something is a jar or the jar is understood from context.

So:

• банка меду can mean a jar of honey or the jar of honey

In this sentence, English would usually translate it as a jar of honey and a jar of sugar, because the sentence is introducing them as things that are in the cupboard.

Yes, it is natural.

The base form is:

• цукор = sugar

But after банка you need the genitive:

• банка цукру = jar of sugar

This is normal Ukrainian grammar. The same idea appears with many other nouns:

• склянка води = glass of water
• пляшка молока = bottle of milk
• тарілка супу = bowl/plate of soup

So цукру is exactly what you would expect here.