После работы я хочу пожарить картофель на ужин.

Because после requires the genitive case.
The noun работа changes to работы in the genitive singular:

• работа = work (dictionary form, nominative)
• после работы = after work

So this is a standard pattern:

• после урока = after the lesson
• после фильма = after the movie
• после работы = after work

In this sentence, после работы is best understood as the fixed everyday expression after work.
Even though работы is grammatically genitive, in natural English you would not usually translate it word-for-word as after the work.

So:

• После работы = After work
• not usually After the work

Because after хотеть (to want), Russian normally uses the infinitive of the action you want to do.

So the structure is:

• я хочу + infinitive
• я хочу читать = I want to read
• я хочу приготовить = I want to cook
• я хочу пожарить = I want to fry

This works very much like English I want to + verb.

This is an aspect question.

• жарить = imperfective
• пожарить = perfective

Here, пожарить is used because the speaker means to fry the potatoes as one complete action, probably for one meal. It sounds like a single planned result: fry them and have them ready for dinner.

Very roughly:

• жарить картофель = to be frying potatoes / to fry potatoes in general
• пожарить картофель = to fry the potatoes (and finish doing it)

After хочу, Russian can use either aspect depending on meaning, but perfective is very common when talking about a specific intended result.

Here картофель is the direct object, so it is in the accusative case.
But because картофель is inanimate masculine singular, its accusative form looks the same as the nominative:

• nominative: картофель
• accusative: картофель

So the form does not change.

As for картошку: that is also very common, and in everyday speech it often sounds more natural.

• картофель = potatoes / potato, more formal or neutral
• картошка = potato / potatoes, more conversational

So these are both possible:

• Я хочу пожарить картофель.
• Я хочу пожарить картошку.

The second one is more common in casual speech.

Grammatically, it is singular, but it often functions like a mass noun in Russian, similar to how English can say potato in a general food sense in some contexts.

In practice, пожарить картофель usually means to fry potatoes as food, not necessarily one single potato.

This is why learners often see both:

• картофель = potatoes/potato as a food product
• картошку = potatoes, very common in everyday speech

На ужин means for dinner. Russian often uses на with meals to express the idea of something being intended for that meal.

Examples:

• на завтрак = for breakfast
• на обед = for lunch
• на ужин = for dinner

So пожарить картофель на ужин means to fry potatoes for dinner.

They are similar, but not identical.

• на ужин = for dinner, as the dinner food
• к ужину = by dinner / for dinner time / ready for dinner

In your sentence, на ужин is the most natural if you mean the potatoes will be eaten as dinner.

Compare:

• Я приготовлю рыбу на ужин. = I’ll cook fish for dinner.
• Я приготовлю рыбу к ужину. = I’ll prepare fish by dinner / in time for dinner.

Sometimes both are possible, but the nuance is slightly different.

Yes. Russian word order is quite flexible because case endings show grammatical roles.

Your sentence is neutral and natural:

• После работы я хочу пожарить картофель на ужин.

But you could also say:

• Я хочу после работы пожарить картофель на ужин.
• Картофель на ужин я хочу пожарить после работы.

These alternatives shift the focus or emphasis, but the basic meaning stays the same.

The original version sounds very normal.

Often, yes. Russian frequently drops subject pronouns when the verb form already makes the subject clear.

So you can say:

• После работы хочу пожарить картофель на ужин.

This still clearly means I want to fry potatoes for dinner after work.

Including я is also correct. It can sound a little more explicit or slightly more emphatic.

It is correct and natural, but in everyday conversation many speakers would probably choose a slightly more colloquial version:

• После работы хочу пожарить картошку на ужин.

Why?

• картошка is more conversational than картофель
• omitting я is common in speech

So your original sentence is good, but it sounds a bit more neutral or slightly more formal because of картофель.

A few useful points:

• после: the о is unstressed and sounds reduced
• работы: stress is on бо → раБОты
• хочу: stress on the last syllable → хоЧУ
• пожарить: stress on жа → поЖАрить
• картофель: stress on то → карТОфель
• ужин: stress on the first syllable → Ужин

A rough stress guide:

• после работы
• я хочу
• пожарить картофель
• на ужин

Getting the stress right will make the sentence sound much more natural.