Moja baka je jednom uspjela ubrati prve tulipane već u ožujku.

Because baka is a feminine singular noun, and the possessive moj has to agree with it.

So:

• moj = masculine singular
• moja = feminine singular
• moje = neuter singular

That is why my grandmother is moja baka.

Croatian words like possessives and adjectives usually agree with the noun in gender, number, and case.

This is the normal Croatian past tense: the perfect (perfekt).

It is formed with:

• a present-tense form of biti (je = is/has as an auxiliary here)
• plus the past participle of the main verb

So:

• je uspjela = managed

The participle uspjela agrees with the subject:

• uspio = masculine singular
• uspjela = feminine singular
• uspjelo = neuter singular

Because baka is feminine, the sentence uses uspjela.

Because uspjeti commonly takes an infinitive after it.

So:

• uspjeti + infinitive = to manage to do something

Examples:

• Uspio je doći. = He managed to come.
• Uspjela je ubrati tulipane. = She managed to pick the tulips.

In this sentence, ubrati is the action she managed to complete.

This is a question of aspect, which is very important in Croatian.

• brati = imperfective: picking, picking repeatedly, picking in general
• ubrati = perfective: pick successfully, complete the picking, pick and obtain the result

Here the sentence describes one successful completed action, so ubrati fits better.

Compare:

• Brala je tulipane. = She was picking tulips / used to pick tulips.
• Ubrala je tulipane. = She picked the tulips (completed action).

After uspjela, the perfective infinitive ubrati sounds very natural because the idea is that she succeeded in completing the action.

Here jednom means once or on one occasion.

So it tells you that this happened one time, not regularly.

It does not mean one in the simple counting sense here. It functions as an adverb.

In different contexts, jednom can also mean things like once, someday, one day, depending on the sentence, but here the meaning is clearly once / one time.

Prve tulipane is in the accusative plural because it is the direct object of ubrati.

She managed to pick what? → prve tulipane

Both words agree:

• prve = adjective
• tulipane = noun

A useful comparison:

• prvi tulipani = the first tulips as a subject
• prve tulipane = the first tulips as a direct object

So the form changes because of the noun’s role in the sentence.

Because Croatian does not have articles like a and the.

So Croatian often expresses definiteness from:

• context
• word order
• adjectives such as prvi (first), taj (that), etc.

In this sentence, prve tulipane naturally means the first tulips, especially in the sense of the first tulips of the season.

English needs an article here, but Croatian does not.

Because Croatian uses u + locative for many time expressions like in March, in summer, in that year, and so on.

Here:

• dictionary form: ožujak = March
• locative form: ožujku

So:

• u ožujku = in March

This is the normal pattern:

• u siječnju = in January
• u travnju = in April
• u prosincu = in December

Već means already, but in this sentence it also gives the feeling of as early as.

So it adds the nuance that March was surprisingly early for that event.

Compare:

• u ožujku = in March
• već u ožujku = already in March / as early as March

Without već, the sentence would still be correct, but it would lose that sense of earliness or unexpectedness.

Croatian word order is fairly flexible, much more flexible than English.

This sentence has a very natural, neutral order:

• Moja baka je jednom uspjela ubrati prve tulipane već u ožujku.

But some parts could move for emphasis. For example:

• Moja baka je već u ožujku jednom uspjela ubrati prve tulipane.
• Jednom je moja baka uspjela ubrati prve tulipane već u ožujku.

These alternatives are possible, but they may sound more marked or shift the emphasis.

In the original sentence:

• jednom smoothly gives the idea once
• već u ožujku is placed late, where it strongly highlights the surprisingly early time

Yes. Uspjeti is normally treated as a perfective verb: it refers to a successful result.

The related imperfective verb is uspijevati.

Compare:

• uspjeti = to manage, to succeed (once, with a result)
• uspijevati = to be managing to, to succeed repeatedly or habitually

Examples:

• Jednom je uspjela. = She succeeded once.
• Često je uspijevala. = She often managed to / was often successful.

Since your sentence describes one completed event in the past, uspjela is exactly the form you would expect.