После ужина осталось немного борща, и утром я разогрел его на обед.

Because the preposition после always requires the genitive.

• ужин = dinner
• после ужина = after dinner

This is a fixed rule: after после, the following noun goes into the genitive.

Because немного means a little / a small amount of, and words of quantity like this normally take the genitive in Russian.

So:

• немного воды = a little water
• немного времени = a little time
• немного борща = a little borscht

Russian is treating борщ here as a substance, so борща means some borscht / a bit of borscht.

This is very common with quantity expressions like немного борща. Russian often uses neuter singular with verbs in sentences of the type there was some X left / there appeared some X / there remained some X.

So:

• осталось немного борща = there was a little borscht left

Even though борщ itself is masculine, the whole quantity expression behaves differently, and the verb is commonly put in neuter singular: осталось.

Compare:

• Остался борщ = The borscht remained / The borscht was left
• Осталось немного борща = There was a little borscht left

Осталось is the perfective past of остаться. It presents the situation as a completed result:

• after dinner, the result was that some borscht was left

That is exactly the natural idea here.

Оставалось would be imperfective and would usually sound more like background, process, or repeated situation:

• После ужина оставалось немного борща could work in a narrative, but it sounds more descriptive or habitual, like after dinner there would usually be a little borscht left or there was still a little left in an ongoing sense.

Because Russian often uses certain time words in the instrumental case as adverbs.

So:

• утром = in the morning
• днём = in the daytime / during the day
• вечером = in the evening
• ночью = at night

This is a normal pattern. There is no need for a preposition like в here.

Разогрел is the perfective form and means warmed up / reheated as a completed action.

• я разогрел его = I reheated it

That fits the sentence well, because the speaker is talking about one finished action.

Грел would be imperfective and would focus more on the process:

• я грел его = I was heating it / I heated it

That is possible in some contexts, but it does not sound as natural here if the main point is simply that the soup was reheated and then eaten.

Because it refers back to борщ, which is a masculine singular noun.

In the second clause, the pronoun is the direct object of разогрел, so it is in the accusative:

• борщ → его = it

For personal pronouns, the accusative form here is его.

An important detail: even though the earlier phrase was немного борща, the pronoun still refers to the thing itself, борщ, so masculine его is used.

Here на обед means for lunch or as lunch.

Russian often uses на + accusative to show purpose or intended use in meal expressions:

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

So:

• я разогрел его на обед = I reheated it for lunch

The idea is that the borscht became the lunch meal.

Yes, Russian word order is fairly flexible. The version in the sentence is natural and neutral:

• После ужина sets the time for the first clause
• утром sets the time for the second clause

So the sentence is organized around time first, which is very common in Russian narrative.

Other orders are possible, for example:

• Утром я его разогрел на обед
• Я утром разогрел его на обед

These can all be correct, but they may shift emphasis slightly. The given order sounds smooth and natural.

Here it is being treated as a substance, like soup in English.

That is why Russian uses:

• немного борща = a little borscht

You are not counting separate items; you are talking about an amount of soup. That is also why the genitive is used after немного.

If you were speaking about a serving or a bowl, Russian would express that differently, for example:

• тарелка борща = a bowl/plate of borscht
• порция борща = a portion of borscht