После магазина нам пришлось докупить хлеб, молоко и яйца.

Because после is a preposition that requires the genitive case.

• магазин = shop/store
• магазина = genitive singular of магазин

So:

• после магазина = after the store / after going to the store

This is just a grammar rule: after после, the next noun goes into the genitive.

Other examples:

• после урока = after the lesson
• после работы = after work
• после обеда = after lunch

Literally, yes, it is after the store, but in natural usage it often means something like:

• after going to the store
• after our trip to the store
• after shopping

Russian often uses short expressions like this where English would use a fuller phrase.

So in this sentence, после магазина does not usually mean standing physically behind a building or something like that. It means after the shopping trip / after leaving the store.

Нам пришлось means we had to.

This comes from the verb прийтись / приходиться, which is often used in an impersonal construction:

• мне пришлось = I had to
• тебе пришлось = you had to
• нам пришлось = we had to

Why нам and not мы?

Because in this structure, the person is put in the dative case:

• мы = we
• нам = to us

So literally, the structure is closer to:

• to us it turned out necessary to...

But in natural English, you should translate it as:

• we had to...

Because пришлось is the past tense neuter singular form used in this impersonal expression.

With пришлось + infinitive, Russian does not agree with нам. The verb stays in this fixed impersonal form:

• мне пришлось ждать = I had to wait
• ему пришлось уйти = he had to leave
• нам пришлось докупить = we had to buy some more

Why not приходилось?

• пришлось usually refers to a specific completed situation in the past
• приходилось often means used to have to or had to repeatedly / under certain circumstances

Compare:

• Нам пришлось докупить хлеб. = We had to buy more bread.
• Нам часто приходилось докупать хлеб. = We often had to buy more bread.

Купить means to buy.

Докупить means to buy additionally, to buy some more, or to buy what was missing.

So:

• купить хлеб = to buy bread
• докупить хлеб = to buy more bread / to buy the remaining bread we still needed

In this sentence, докупить suggests that some shopping had already been done, but then they realized they still needed bread, milk, and eggs.

That is why докупить is a very natural choice with после магазина: they already went shopping, but afterward they had to go back or make an extra purchase.

Because after пришлось, Russian normally uses an infinitive:

• пришлось + infinitive

Examples:

• Мне пришлось уйти. = I had to leave.
• Ей пришлось работать. = She had to work.
• Нам пришлось докупить хлеб. = We had to buy some more bread.

So the pattern is:

• [dative person] + пришлось + infinitive

That is a very common Russian structure.

They are the direct objects of докупить, so they are in the accusative case.

However, for these nouns, the accusative looks the same as the nominative:

• хлеб → хлеб
• молоко → молоко
• яйца → яйца

Why?

• Inanimate masculine nouns often have accusative = nominative in the singular.
• Neuter nouns also often have accusative = nominative.
• Inanimate plural nouns usually have accusative = nominative.

So even though these words are objects, their forms do not visibly change here.

Because Russian treats these nouns differently, just like English does.

• хлеб = bread, usually treated as an uncountable mass noun
• молоко = milk, also usually uncountable
• яйца = eggs, normally countable, so plural is natural

So the sentence is basically listing shopping items in the most natural form:

• bread
• milk
• eggs

This is very similar to English.

Yes, the word order could be changed, because Russian word order is more flexible than English word order.

The given sentence:

• После магазина нам пришлось докупить хлеб, молоко и яйца.

is a natural neutral way to say it.

You could also say:

• Нам пришлось после магазина докупить хлеб, молоко и яйца.
• Хлеб, молоко и яйца нам пришлось докупить после магазина.

But the emphasis changes.

The original version starts with После магазина, which sets the scene first:

• After the shopping trip / after the store, we had to...

That sounds very natural.

Very often, yes.

Because of докупить, the sentence strongly suggests:

• they had already done some shopping
• then they realized something was missing
• so they had to buy additional items

So the overall feeling is something like:

• After going to the store, we still had to buy more bread, milk, and eggs.

It does not necessarily mean they completely forgot them, but it does suggest that the original shopping was not enough or was incomplete.

Sometimes, but usually had to is better.

Пришлось often expresses necessity because of circumstances, not necessarily strong force or pressure.

So:

• Нам пришлось докупить хлеб. = We had to buy more bread.

We were forced to buy more bread sounds much stronger in English and may suggest some outside pressure. Russian пришлось is often more neutral:

• it turned out to be necessary
• circumstances made it necessary

So had to is usually the best translation.