Ben akşam yemeği için patatesleri soyuyorum.

Yes. Turkish often drops subject pronouns when the verb already shows who is doing the action.

• Ben soyuyorum = I am peeling
• Soyuyorum by itself already means I am peeling

So Ben akşam yemeği için patatesleri soyuyorum and Akşam yemeği için patatesleri soyuyorum both work.
Including ben can add emphasis, contrast, or clarity, such as I’m the one peeling the potatoes.

It means for dinner or more literally for the evening meal.

Breakdown:

• akşam = evening
• yemek = food / meal / to eat
• akşam yemeği = dinner, evening meal
• için = for

So:

• akşam yemeği için = for dinner

This phrase shows purpose: the potatoes are being peeled for dinner.

In akşam yemeği, the word yemeği is part of a Turkish noun phrase construction that often works like X meal, X door, X color, and so on.

Here:

• akşam modifies yemek
• yemek takes the 3rd-person possessive ending and becomes yemeği

So akşam yemeği is the normal way to say dinner.
It is not literally saying his/her dinner in the usual sense here; it is just the standard compound-like structure.

Because the potatoes are the definite direct object of the verb soymak (to peel), so Turkish uses the accusative ending.

Breakdown:

• patates = potato
• patatesler = potatoes
• patatesleri = the potatoes

In Turkish, a specific or known direct object usually takes the accusative:

• Patates soyuyorum = I am peeling potatoes / some potatoes
• Patatesleri soyuyorum = I am peeling the potatoes

So patatesleri suggests specific potatoes, not potatoes in general.

In some contexts, patatesleri can also mean his/her potatoes or their potatoes, because the form can overlap.

Possible meanings depending on context:

• the potatoes
• his/her potatoes
• their potatoes

But in this sentence, the most natural meaning is the potatoes, because it is functioning as a definite direct object with soyuyorum.

Soyuyorum is in the present continuous form, often used like I am peeling.

Breakdown:

• soy- = peel
• -uyor- = present continuous marker
• -um = I

So:

• soyuyorum = I am peeling

This form can also sometimes be used for near-future meaning depending on context, but here it most naturally means an action happening now.

The present continuous suffix changes form because of vowel harmony and sound patterns.

The basic suffix is often shown as:

• -iyor / -ıyor / -uyor / -üyor

Which version appears depends on the vowel in the verb stem.

Here:

• soymak has the vowel o
• so the suffix becomes -uyor
• soy + uyor → soyuyor
• then -um is added: soyuyorum

So soyuyorum is the correct harmonized form.

The dictionary form is soymak, meaning to peel.

Many Turkish verbs are listed with -mak or -mek:

• soymak = to peel
• yemek = to eat
• gitmek = to go

To build the sentence, Turkish removes -mak and adds tense/person endings:

• soymak → soy-
• soy- + uyor + um → soyuyorum

Turkish word order is flexible, but the most neutral pattern is often:

subject + time/purpose/etc. + object + verb

So:

• Ben = subject
• akşam yemeği için = purpose phrase
• patatesleri = object
• soyuyorum = verb

The verb usually comes at the end in a neutral Turkish sentence.

English prefers:

• I am peeling the potatoes for dinner

Turkish prefers:

• I for dinner the potatoes am peeling

That is why the Turkish sentence feels reversed compared with English.

Yes, very natural.

A common version would be:

• Akşam yemeği için patatesleri soyuyorum.

This is probably what many native speakers would say in normal conversation unless they want to stress I.

Not with the same meaning.

• akşam yemeği için = for dinner
• akşam yemeğine usually means to dinner or for the dinner event/meal depending on context

So in this sentence, için is the correct choice because it expresses purpose: the potatoes are being peeled for dinner.

It would sound less specific.

Compare:

• Patates soyuyorum = I am peeling potatoes
• Patatesleri soyuyorum = I am peeling the potatoes

The version without the accusative ending is more indefinite or generic.
The version with -leri here points to particular potatoes already understood in the situation.

Usually soyuyorum most strongly suggests I am peeling right now or around now.

For a habitual meaning, Turkish often prefers the aorist:

• Ben akşam yemeği için patatesleri soyarım.
• This would mean something like I peel the potatoes for dinner / I usually peel the potatoes for dinner

So the given sentence is mainly understood as a present ongoing action.