Prošle godine su mi već jednom odbili zahtjev, pa sam tada pisala žalbu.

Prošle godine is a very common “time when” expression meaning last year. Grammatically it’s a genitive form (godine is genitive singular of godina) used adverbially with time expressions (similar to ovog tjedna = this week, sljedeće godine = next year).
You can say u prošloj godini, but it sounds more formal/abstract (more like “in the previous year” in a reporting/statistical sense).

Su is the present-tense auxiliary to be used to form the past tense (Croatian “perfect”): odbili su = they refused/they have refused.
Croatian has a strong tendency for clitics (short unstressed words like su, sam, mi) to appear in the “second position” of the clause. That’s why Prošle godine su… is natural.

Croatian often omits the subject when it’s understood from context or when it’s an impersonal “they” (an institution/office/people in charge). The verb form odbili is plural, so it implies they without naming them.

Mi means to me and it’s dative singular (1st person).
In this sentence it marks the person affected by the action: they refused the request to me / in my case. With verbs like odbiti (to refuse), it’s common to say odbiti nekome nešto = refuse someone something.

This is clitic ordering. In Croatian, clitics cluster near the beginning of the clause in a fairly fixed internal order. Very roughly: 1) auxiliary (su/sam/je…)
2) dative/accusative pronoun clitics (mi/te/mu/ga…)
Then other words (like već, jednom) and finally the main verb participle (odbili).
You can move the participle: Prošle godine odbili su mi… is also possible, but the clitics still stick together: odbili su mi, not mi su.

Jednom literally means once (originally an instrumental form), and već adds the nuance already / as it happened before.
So već jednom is like already once / once before (already), emphasizing that it wasn’t the first time.

Odbili is the past participle (masculine plural) from the verb odbiti = to refuse/decline. It’s plural because the implied subject is plural (they = officials/office/people).
If the subject were singular masculine: odbio je; singular feminine: odbila je; neuter: odbilo je.

Zahtjev is accusative singular here because it’s the direct object of odbiti (to refuse something). For masculine inanimate nouns like zahtjev, accusative singular looks the same as nominative singular, so it doesn’t change form.

Pa commonly links two clauses with a “result/continuation” feel: so / and then / and therefore depending on context. It often implies: “given that, the next thing happened.”

• i is a more neutral and.
• onda is more purely temporal then.
  Pa here nicely signals: refusal happened → as a consequence, the next action followed.

Sam is the auxiliary “am” used to form the past tense with pisala (I wrote/was writing). Like su, it’s a clitic and tends to appear in second position: pa sam tada…

Past participles agree with the subject in gender and number. Pisala is feminine singular, so the speaker is female (or speaking in a feminine grammatical role).
A male speaker would say: pa sam tada pisao žalbu.

Pisala (from pisati, imperfective) can mean wrote in the general sense or can highlight the process/habitual nature (“I was writing / I wrote (as an activity)”).
If you want to stress a completed, one-off result (finished letter/complaint), Croatian often uses the perfective: napisala sam žalbu (“I wrote it / I wrote up a complaint”).

Žalbu is accusative singular of žalba (a complaint/appeal). It’s the direct object of pisati (to write). Feminine -a nouns typically change -a → -u in the accusative singular (žalba → žalbu).