Мне пришлось подняться на восьмой этаж по лестнице.

Because пришлось is used in an impersonal construction.

Russian says this literally more like:

• Мне пришлось... = It became necessary for me... / I had to...

So the person affected goes into the dative case:

• мне = to me / for me
• тебе = to you
• ему = to him

You cannot say я пришлось, because пришлось does not take a normal nominative subject in this pattern.

Also, пришлось is a bit different from я должен был:

• мне пришлось = circumstances forced it; I ended up having to do it
• я должен был = I was supposed to / I was obliged to
• мне надо было = it was necessary for me; more neutral

So мне пришлось подняться... often suggests something like I had no real choice; I had to go up...

Пришлось is the past tense of прийтись in this impersonal meaning: to have to, to be forced to, to end up needing to.

In this sentence, it means:

• I had to go up to the eighth floor by the stairs

But the nuance is often stronger than plain English had to. It can imply:

• because of the situation
• unwillingly or inconveniently
• not originally planned

For example, this sentence could imply something like the elevator was broken.

Because impersonal past-tense constructions in Russian normally use the neuter singular form.

So you get:

• мне пришлось
• нам пришлось
• ей пришлось

Even though the real-life person is I, we, or she, the verb still stays in this impersonal neuter form.

This is one of the standard patterns of Russian grammar, not something special to this one sentence.

Because подняться is perfective, and here the action is viewed as a single completed event:

• I had to go up / make it up to the eighth floor

The focus is on reaching the destination.

Compare:

• Мне пришлось подняться на восьмой этаж
  = I had to go up to the eighth floor.
  Focus: completed result.

• Мне пришлось подниматься на восьмой этаж пешком
  = I had to be climbing / had to go up to the eighth floor on foot.
  Focus: process, effort, duration, or repeated motion.

So in your sentence, подняться is natural because the destination matters.

Here -ся is part of the verb подняться, which means to rise / to go up / to make one’s way up.

Compare:

• поднять = to lift something
• подняться = to go up / rise oneself

So:

• поднять чемодан = to lift a suitcase
• подняться на этаж = to go up to a floor

In this sentence, подняться is not really translated as a reflexive oneself in English. It is simply the normal Russian verb for go up in this context.

Because Russian uses:

• на + accusative for motion to a floor
• на + prepositional for location on a floor

So:

• на восьмой этаж = to the eighth floor
• на восьмом этаже = on the eighth floor

This is a very common Russian pattern with floors:

• Я живу на пятом этаже. = I live on the fifth floor.
• Я поднялся на пятый этаж. = I went up to the fifth floor.

English learners often expect в, but with floors Russian normally uses на.

Yes, it is accusative here, but the form happens to look the same as the nominative.

Why? Because:

• этаж is masculine
• it is inanimate
• for masculine inanimate nouns, the accusative usually matches the nominative
• the adjective/ordinal matches that pattern too

So:

• nominative: восьмой этаж
• accusative: на восьмой этаж

Same form, different function.

If it were location, you would see a different case:

• на восьмом этаже

По лестнице uses the preposition по with the dative case.

So:

• лестница = staircase / stairs
• по лестнице = by the stairs / along the staircase / using the stairs

A very important point: лестнице here is dative singular, not instrumental.

Forms:

• nominative: лестница
• dative: лестнице
• instrumental: лестницей

So the sentence is по лестнице, not лестницей.

In this context, по лестнице means the route or means of movement: via the stairs.

Because Russian often uses лестница as a singular noun where English prefers the plural stairs.

So:

• по лестнице = by the stairs / up the stairs
• на лестнице = on the stairs

This is normal Russian usage. English and Russian just divide up the idea differently:

• English: stairs is usually plural
• Russian: лестница is often singular for one staircase/stairway

Often, yes.

By itself, по лестнице simply means by the stairs / using the stairs. But in many real situations it strongly suggests a contrast such as:

• not by elevator
• not by escalator
• not by some easier method

So the sentence can carry the feeling:

• I had to go up to the eighth floor by the stairs
  maybe because the elevator wasn’t working

That contrast is often understood from context, even if the elevator is not mentioned.

Yes. Russian word order is flexible.

This sentence is a very natural order:

• Мне пришлось подняться на восьмой этаж по лестнице.

But Russian can rearrange parts for emphasis:

• Подняться на восьмой этаж мне пришлось по лестнице.
• По лестнице мне пришлось подняться на восьмой этаж.

The basic meaning stays the same, but the emphasis changes.

In your original sentence, starting with мне пришлось sounds natural because it introduces the main idea first: I had to...

Yes, but it is not exactly the same.

Compare:

• Мне пришлось подняться на восьмой этаж по лестнице.
  = I had to go up to the eighth floor by the stairs.
  Strong sense of upward movement and reaching the floor.

• Мне пришлось идти на восьмой этаж по лестнице.
  = I had to go to the eighth floor by the stairs.
  Grammatically possible, but less focused on the upward motion itself.

Because going to a higher floor naturally involves upward movement, подняться is usually the more precise and idiomatic choice.

Yes, very naturally.

Two parts help create that feeling:

• пришлось = had to / was forced to
• на восьмой этаж по лестнице = to the eighth floor by the stairs

So even without extra words, the sentence can sound like:

• it was inconvenient
• it was tiring
• the speaker may be mildly complaining

Russian often conveys this kind of attitude through word choice rather than an explicit comment like unfortunately or annoyingly.