Мне удалось вовремя добраться до вокзала, несмотря на пробку.

Мне удалось is a very common Russian pattern meaning I managed / I succeeded (in doing something).
Grammatically it’s an “impersonal” construction: the verb удаться (“to succeed, to turn out well”) doesn’t take a normal subject like я. Instead, the person is shown in the dative: мне = “to me / for me.”
It often implies “it worked out (for me)”—sometimes a bit of luck or circumstances helping.
Я смог(ла) = “I was able to” is more neutral and ability-focused.

удалось is:

• past tense
• perfective aspect (completed success)
• neuter singular past form

It’s neuter because there’s no real grammatical subject like я; it behaves like “it” in English: “It succeeded (for me).”

Because this structure expresses success for someone rather than an action done by a subject.
So Russian marks the person affected/benefiting with the dative:

• мне удалось = “I managed” (literally: “it succeeded for me”)
• ему удалось = “he managed”
• нам удалось = “we managed”

After удаться, Russian normally uses an infinitive to say what you managed to do:

• Мне удалось добраться... = “I managed to get...” It’s similar to English “manage to + verb.”

добраться means to get (to a place), to make it there, often emphasizing difficulty, distance, or obstacles.
The -ся here is part of the verb’s standard form; it doesn’t mean “myself” in a literal way. Many motion/achievement verbs are reflexive in Russian with this meaning:

• добраться до дома = “get home”
• добраться до станции = “reach the station”

Different prepositions express different nuances:

• добраться до + Genitive = to make it as far as / reach (goal achieved)
  добраться до вокзала = “reach the station”
• в + Accusative = “into” (entering a building/space)
  в вокзал would sound like “into the station building,” and it’s not the typical pair with добраться
• к + Dative = “to/toward, up to (near)” (arrival near, not necessarily “reach” in the sense of overcoming difficulty)
  до is the standard with добраться

Because до requires the genitive case:

• до вокзала, до дома, до работы, etc.

вовремя means on time / in time (i.e., not late).
Its position is flexible, but it commonly appears near the verb phrase:

• Мне удалось вовремя добраться... (very natural) You could also say:
• Мне удалось добраться до вокзала вовремя. The meaning stays basically the same; word order mostly changes emphasis.

несмотря на means despite / in spite of and it requires the accusative case:

• несмотря на пробку (traffic jam = пробка, accusative пробку)
• несмотря на дождь
• несмотря на трудности

Because на in the fixed expression несмотря на takes accusative:

• nominative: пробка
• accusative: пробку

Yes, common alternatives include:

• вопреки пробке = “contrary to the traffic jam” (more formal; вопреки + dative)
• даже с пробкой / даже при пробке = “even with a traffic jam” (different nuance)
• хотя была пробка = “although there was a traffic jam” (full clause)

But несмотря на пробку is the most straightforward “despite.”

Yes, that’s very natural. You can front the “despite” phrase for emphasis:

• Несмотря на пробку, мне удалось вовремя добраться до вокзала. Both versions are correct; the fronted version highlights the obstacle first.