После экзамена мне было не до прогулок, и я сразу пошёл домой.

Because после always takes the genitive.

So:

• после экзамена = after the exam
• not после экзамен

A few more examples:

• после урока = after the lesson
• после работы = after work
• после дождя = after the rain

So this is simply a preposition + case rule: после + genitive.

This is an impersonal construction, which is very common in Russian.

Literally, мне было не до прогулок is something like:

• to me it was not for walks

But in natural English it means:

• I wasn't up for a walk
• I didn't feel like taking a walk
• I had no time/energy for walks

Russian often uses the dative to show the person experiencing a state:

• мне холодно = I am cold
• мне грустно = I am sad
• мне было не до шуток = I was not in the mood for jokes

So мне does not mean to me in a fully literal English sense here; it marks the person affected by the situation.

Не до + genitive is an idiomatic expression meaning:

• not up to
• not in the mood for
• not in a position to deal with
• having no time/energy for

So мне было не до прогулок means that after the exam, the speaker was too tired, stressed, distracted, or preoccupied to think about going for a walk.

This expression is very common:

• Мне сейчас не до разговоров. = I'm not up for talking right now.
• Ему было не до смеха. = He was in no mood for laughter.
• Нам не до отдыха. = We have no time for rest.

It often suggests that something more important or exhausting is going on.

Because не до requires the genitive case, and прогулок is the genitive plural form of прогулки.

Base form:

• прогулка = a walk

Forms:

• прогулка = nominative singular
• прогулки = nominative plural / genitive singular
• прогулок = genitive plural

Since the phrase is не до прогулок, the noun must be in the genitive.

The plural here sounds general and idiomatic: walks / going for walks / any kind of walk.

So не до прогулок means something like:

• not up for walks
• not up for going out for a walk
• not in the mood for that kind of thing

Using the plural often gives a broad, non-specific sense.

A singular form could exist in a very specific context, but не до прогулок is the natural everyday phrasing when you mean walking as an activity in general.

Compare:

• Мне не до прогулок. = I’m not up for walks / going out.
• Мне не до шуток. = I’m not in the mood for jokes.
• Им не до разговоров. = They’re not up for talking.

This sentence is impersonal, so there is no normal subject like я controlling the verb.

In Russian, impersonal past-tense predicates often use neuter singular:

• было
• стало
• оказалось

So:

• мне было не до прогулок = literally an impersonal structure, not I was in the normal subject-verb sense

This is similar to other Russian impersonal sentences:

• Мне было холодно. = I was cold.
• Ему было трудно. = It was hard for him / He found it difficult.
• Нам было весело. = We were having fun.

So было is not agreeing with мне. It is simply the standard past-tense form used in this impersonal pattern.

Пошёл is the past tense of пойти, a perfective verb. Here it means:

• went
• set off
• headed

It presents the action as a single completed event: after the exam, the speaker immediately went home.

Шёл is from идти, an imperfective verb, and would focus more on the process:

• я шёл домой = I was walking home / I went home on foot

In this sentence, the important idea is the decision/result:

• after the exam, I immediately went home

So пошёл домой is the natural choice.

Because домой means to home / homeward, showing motion toward home.

Compare:

• я пошёл домой = I went home
• я был дома = I was at home

So:

• домой = destination
• дома = location

This is a very important distinction in Russian.

Usually no, not if you mean home in the normal English sense.

• домой = home, to one’s home
• к дому = toward the house / up to the house

К дому refers more literally to a building as an object, not to the idea of going home as your destination in the usual personal sense.

So:

• Я пошёл домой. = I went home.
• Я подошёл к дому. = I walked up to the house.

Those are different meanings.

It is not strictly necessary, because пошёл already shows:

• past tense
• masculine singular

So in context, пошёл could be understood without я.

However, Russian often includes the pronoun for clarity, rhythm, or emphasis, especially in a compound sentence like this one. Here я helps clearly mark the second clause:

• После экзамена мне было не до прогулок, и я сразу пошёл домой.

Without я, the sentence would still be possible:

• После экзамена мне было не до прогулок, и сразу пошёл домой.

But that version sounds less natural by itself because the subject is not explicitly restated after the conjunction.

Because и is joining two separate clauses:

1. После экзамена мне было не до прогулок
2. я сразу пошёл домой

Each part has its own predicate:

• было
• пошёл

So a comma is normally used before и when it connects two independent clauses.

Сразу means:

• immediately
• right away
• straight away

So я сразу пошёл домой means the speaker did not stay around, chat, relax, or go for a walk. He went home at once.

It strengthens the overall idea of the sentence:

• the exam was tiring or stressful
• he was not up for a walk
• so he went straight home

Yes. Мне было не до прогулок does not only mean physical tiredness. It can suggest several things, such as:

• mental exhaustion
• stress
• worry
• lack of time
• lack of desire
• being preoccupied with something else

So after an exam, the speaker might be:

• tired
• emotionally drained
• anxious about the result
• simply wanting to get home as soon as possible

That is why не до is such a useful expression: it covers a broader feeling than just I was tired.