Без электричества лифт не работал, поэтому я поднялась по лестнице пешком.

The preposition без (without) normally requires the genitive case.
So электричество (nominative) → электричества (genitive): без электричества = without electricity.

It’s a prepositional phrase, not a full clause. Russian often uses this kind of “setting phrase” to give the circumstance:
Без электричества = With no electricity / Without electricity.
You could also say a fuller version like Не было электричества, и лифт не работал... (There was no electricity, and the elevator didn’t work...), but the given sentence is completely natural as-is.

Past tense in Russian agrees in gender and number with the subject.
лифт is masculine singular, so the verb is работал (masculine past).

• masculine: работал
• feminine: работала
• neuter: работало
• plural: работали

не работал (imperfective) means wasn’t working / didn’t work (over a period of time)—a general state.
не сработал (perfective) is more like didn’t work (at one moment) / failed to activate—a one-time failure.
Here, не работал fits because the elevator simply wasn’t functioning due to the outage.

поэтому means therefore / so / that’s why, introducing a result.
A comma is used because it separates two parts of a complex sentence:
[cause], поэтому [result].
In other words: The elevator didn’t work, therefore I walked up the stairs.

Yes, often.

• поэтому = therefore / that’s why (slightly more “logical/explicit”)
• так что = so / so that / as a result (often more conversational)

Example: ...лифт не работал, так что я поднялась... is also natural.

поднялась (perfective) focuses on a completed result: you ended up at the top.
поднималась (imperfective) focuses on the process: you were walking up (without emphasizing completion).
In this context, the “result” reading is most typical: she got upstairs.

Two things are happening: 1) Gender agreement in the past: the speaker is presumably female, so past tense is feminine: подняла- + -сь
2) -сь is the reduced form of -ся, marking a reflexive verb: подняться = to go up / rise (oneself).

So я поднялась = I (female) went up.

Only the past-tense gender marking:

• female: я поднялась
• male: я поднялся
  Everything else can stay the same.

After по meaning along / via / by (a route), Russian usually uses the dative case.
So лестница → лестнице (dative): по лестнице = by the stairs / via the staircase / up the stairs.

пешком means on foot and emphasizes that you didn’t take any vehicle/transport (and in some contexts contrasts with taking an elevator/escalator).
It can sound slightly redundant in English, but in Russian it’s a common, natural intensifier: I went up the stairs on foot (i.e., not by elevator).

Russian word order is flexible, but it changes emphasis.

• Без электричества лифт не работал emphasizes the condition first (Without electricity...)
• Лифт не работал без электричества is possible but can sound like “The elevator doesn’t work without electricity (in general),” more like a general statement.

For this specific situation (power outage), the original order is more natural.