Prošle zime bila sam na bolovanju tjedan dana zbog gripe.

Because Croatian often leaves out subject pronouns when they are already clear from the verb form. In bila sam, the auxiliary sam already shows first person singular, so ja is not necessary.

You could add ja for emphasis or contrast:

• Ja sam bila na bolovanju, a on nije. = I was on sick leave, but he wasn’t.

It is the Croatian perfect tense, formed with:

• the auxiliary sam = am
• the past participle bila

So bila sam literally corresponds to I was / I have been, but in this sentence English normally uses the simple past: I was.

Because the past participle agrees with the speaker’s gender.

• A woman says bila sam
• A man says bio sam

So this sentence is spoken by a female speaker.

Because prošle zime is a time expression meaning last winter. Croatian commonly uses the genitive for expressions like this.

Base form:

• prošla zima = last winter as a dictionary-style or subject form

But in a temporal phrase:

• prošle zime = last winter / during last winter

Compare:

• Prošla zima bila je blaga. = Last winter was mild.
  Here prošla zima is the subject.
• Prošle zime bila sam na bolovanju. = Last winter I was on sick leave.
  Here it is a time phrase.

Because the adjective has to agree with zime in gender, number, and case.

The noun zima is:

• feminine singular in the base form

In this sentence it becomes genitive singular:

• zime

So the adjective changes too:

• prošla zima → prošle zime

It means on sick leave.

This is an idiomatic expression in Croatian. It usually means someone is officially away from work, and sometimes school, because of illness.

So it is more specific than just being sick:

• bolesna sam = I’m sick
• na bolovanju sam = I’m on sick leave

It is locative singular of bolovanje.

The phrase na bolovanju uses na + locative, because that is the fixed Croatian expression for on sick leave.

Similar expressions:

• na poslu = at work
• na odmoru = on vacation
• na fakultetu = at university

So this is best learned as a whole chunk: biti na bolovanju.

Because Croatian often expresses duration with the accusative, without any preposition.

So:

• tjedan dana = for a week
• dva sata = for two hours
• cijelu noć = all night / for the whole night

This is important because za tjedan dana does not mean for a week. It usually means in a week.

Tjedan dana is the normal Croatian way to say a week when talking about duration.

Literally, it is something like a week of days:

• tjedan = week
• dana = genitive plural of dan (day)

This pattern is very common:

• mjesec dana = a month
• godinu dana = a year
• par dana = a few days

You may also hear jedan tjedan, but tjedan dana is usually more natural here.

It is the accusative singular, used to express duration.

For masculine inanimate nouns like tjedan, the accusative singular looks the same as the nominative singular, so the form does not change:

• nominative: tjedan
• accusative: tjedan

That is why it may look like a dictionary form even though it is functioning as an object of duration.

Because zbog requires the genitive case.

The noun is:

• gripa = flu

Its genitive singular is:

• gripe

So:

• zbog gripe = because of the flu

The same pattern appears in many phrases:

• zbog kiše = because of the rain
• zbog posla = because of work
• zbog bolesti = because of illness

Yes. Croatian word order is fairly flexible.

Both of these are possible:

• Prošle zime bila sam na bolovanju tjedan dana zbog gripe.
• Prošle zime sam bila na bolovanju tjedan dana zbog gripe.

The meaning stays the same. Many speakers would find Prošle zime sam bila... a little more neutral in everyday speech, but the version with bila sam is also natural. Word order in Croatian often changes for rhythm, emphasis, or style.