Зимой сухость кожи на руках не проходит быстро.

A good stress-marked version is:

Зимóй сýхость кóжи на рукáх не прохóдит бы́стро.

A few pronunciation notes:

• зимой = zi-MOY
• сухость = SU-khastʹ
• кожи = KO-zhi
• руках = ru-KAKH
• проходит = pra-KHO-dit
• быстро = BY-stra

Also:

• х sounds like the ch in Scottish loch or German Bach
• ы in быстро is the hard Russian vowel that has no exact English equivalent

Зимой is the instrumental singular of зима.

In Russian, seasons are very often used in the instrumental case without a preposition to mean in/during that season:

• зимой = in winter
• летом = in summer
• осенью = in autumn
• весной = in spring

So зимой here means in winter or during the winter.

Using в зимой would be incorrect.
Using зима would just be the noun winter, not the time expression needed here.

Both are possible in Russian, but they are not exactly the same.

• сухая кожа = dry skin
• сухость кожи = dryness of the skin

So сухость is a noun meaning dryness, while сухая is an adjective meaning dry.

The sentence is talking about the condition/symptom itself, so сухость кожи sounds slightly more medical, descriptive, or formal. It is very natural in skincare, health, or product-related language.

Because Russian often uses the pattern noun + genitive to show what something belongs to, what it concerns, or what kind of thing it is.

Here:

• сухость = dryness
• кожи = of the skin

So:

• сухость кожи = the dryness of the skin

This is a very common pattern:

• цвет неба = the color of the sky
• запах кофе = the smell of coffee
• конец фильма = the end of the film

The dictionary form is кожа, but after сухость it becomes кожи.

На руках means on the hands.

Here на is used because the skin is located on the surface of the hands. Russian commonly uses на for that kind of location.

After на, when it means location, Russian uses the prepositional case:

• singular: на руке = on the hand
• plural: на руках = on the hands

So руках is the prepositional plural of руки.

Compare:

• на руках = on the hands
• в руках = in the hands

Those mean different things.

Because they are referring to different kinds of things.

• руках is plural because people usually mean both hands in a general sense
• кожи is singular because skin is treated as a mass noun, just like in English

So Russian naturally says:

• кожа на руках = skin on the hands

Even though there are two hands, the noun skin stays singular because it refers to the substance or tissue in general.

Yes, literally проходить can mean things like to pass, to go by, or to go through. But with symptoms, pain, illnesses, and conditions, it often means:

• to go away
• to subside
• to clear up

So here:

• сухость кожи не проходит = the skin dryness does not go away

This is very common in Russian:

• боль проходит = the pain is going away
• простуда проходит = the cold is going away
• сыпь не проходит = the rash is not going away

In Russian, the present tense of an imperfective verb is often used for a general fact, habitual situation, or usual pattern.

So не проходит быстро here means something like:

• doesn’t go away quickly
• doesn’t clear up quickly
• is slow to go away

It is not necessarily talking about one single moment right now. It describes what generally happens in winter.

If you wanted to talk about one specific future case, Russian might use:

• не пройдёт быстро = it won’t go away quickly

So the present tense here is very natural for a general statement.

Because the sentence describes an ongoing or typical process, not a single completed result.

• проходить = imperfective
• пройти = perfective

Imperfective is used when Russian focuses on:

• the process
• repeated or usual situations
• general facts

That is exactly what is happening here: in winter, this condition does not go away quickly as a general rule.

If you said не пройдёт быстро, that would sound more like:

• it will not go away quickly
• this specific case will not clear up quickly

So не проходит is the better choice for a broad statement.

Yes, the word order is natural.

Russian word order is flexible, but this version is neutral and clear:

• Зимой сухость кожи на руках не проходит быстро.

Putting быстро at the end sounds normal and keeps the final idea on quickly.

You could also say:

• Зимой сухость кожи на руках быстро не проходит.

That is also possible, but it puts a little more emphasis on быстро, almost like:

• it doesn’t go away quickly
• quickly is not how it goes away

So the original order is a good neutral choice.

It is natural and correct, but сухость кожи sounds a little more formal or clinical than everyday speech.

That makes the sentence especially suitable for:

• skincare
• medicine
• pharmacy
• health advice
• product descriptions

In everyday conversation, someone might also say:

• Зимой кожа на руках долго остаётся сухой.
• Зимой кожа на руках сохнет и долго не проходит.
• Зимой сухость на руках долго не проходит.

So the original sentence is perfectly good, but it has a slightly more descriptive or medical tone.