On je ovaj tjedan zauzet projektom, ali vikendom ima pravo na odmor.

Yes, you can omit On. Croatian is a pro‑drop language, so the verb form usually tells you who the subject is.

• On je ovaj tjedan zauzet projektom…
• Je ovaj tjedan zauzet projektom… – grammatically possible, but sounds incomplete without any subject or context
• Ovaj tjedan je zauzet projektom… – perfectly natural; on is understood

We normally include On:

• for emphasis (contrast: On is busy, someone else is not)
• when introducing a person for the first time
• to avoid ambiguity when context is unclear

In many everyday contexts, you would more likely hear:
Ovaj tjedan je zauzet projektom, ali vikendom ima pravo na odmor.

In Croatian, a bare time expression like ovaj tjedan often functions on its own as a time adverbial, without a preposition.

• ovaj tjedan = this week (as an adverbial: this week he is busy…)
• sljedeći tjedan = next week
• prošle godine = last year

You can say u ovom tjednu, but that sounds more formal or abstract, often about time as a block, e.g.:

• U ovom tjednu imamo tri sastanka.
  In this week we have three meetings.

In everyday speech for “this week”, ovaj tjedan without a preposition is the normal choice.

Zauzet is the masculine singular past participle of the verb zauzeti (to occupy, to take, to engage), but here it functions as an adjective meaning busy / occupied.

Structure:

• On je zauzet. – He is busy.
• On je zauzet projektom. – He is busy with the project.

So:

• je = present tense of biti (to be)
• zauzet = adjective (originally a participle)

Together, they make a present‑time description, similar to English “is busy”, not a past tense.

Projektom is in the instrumental case. With the adjective zauzet, Croatian uses the instrumental to express “busy with something”.

Pattern:

• zauzet + instrumental
  • zauzet poslom – busy with work
  • zauzet učenjem – busy with studying
  • zauzet projektom – busy with the project

Instrumental often answers čime? (with what?).
So čime je zauzet? – (On je) zauzet projektom.

Vikendom is the instrumental singular of vikend (weekend), used adverbially to express a habitual time: on weekends / at weekends.

Compare:

• vikend (nom.) – Vikend je kratak. – The weekend is short.
• vikendom (instr.) – Vikendom spavam duže. – On weekends I sleep longer.

You will also hear:

• za vikend – literally for the weekend, often used for a specific upcoming weekend
  
  • Za vikend idem na more. – I’m going to the sea this weekend.
• vikendima – on weekends (plural form, less common in some regions)

In your sentence, vikendom = on weekends (generally, as a habit).

Ima pravo na odmor is an idiomatic phrase meaning “has the right to rest / is entitled to rest.”

• ima odmor = has a break / has time off (more concrete: he actually has vacation or a break scheduled)
• ima pravo na odmor = has the right to rest (legal, moral, or general entitlement, regardless of whether he actually takes it)

The pattern is:

• imati pravo na + accusative
  • ima pravo na odmor – has the right to rest
  • ima pravo na mišljenje – has the right to an opinion
  • ima pravo na žalbu – has the right to appeal

The preposition na is required by the fixed expression imati pravo na + accusative (to have the right to something).

So:

• pravo na odmor – the right to rest
• pravo na obrazovanje – the right to education
• pravo na život – the right to life

You can’t replace na with another preposition in this phrase.
*pravo za odmor or *pravo od odmora are incorrect in standard Croatian in this sense.

In Croatian, the present tense is regularly used for:

• general truths
• permanent or habitual situations

Examples:

• Čovjek ima pravo na slobodu. – A person has a right to freedom.
• Djeca imaju pravo na obrazovanje. – Children have the right to education.

Similarly:

• …ali vikendom ima pravo na odmor.
  …but on weekends he has the right to rest.

No extra tense or aspect is needed. Croatian doesn’t have a separate “simple present” vs “present continuous” contrast like English; the same present form covers both.

Yes, Croatian word order is relatively flexible. All of these are grammatically correct, with slight differences in emphasis:

• On je ovaj tjedan zauzet projektom, ali vikendom ima pravo na odmor.
  (Neutral, straightforward order.)

• Ovaj tjedan je on zauzet projektom, ali vikendom ima pravo na odmor.
  (Emphasises this week and he in contrast to others.)

• Ovaj tjedan je zauzet projektom, ali vikendom ima pravo na odmor.
  (Very natural; subject on is understood from context.)

• On je zauzet projektom ovaj tjedan, ali vikendom ima pravo na odmor.
  (More emphasis at the end on ovaj tjedan, like English “He’s busy with the project this week.”)

Meaning stays essentially the same; movement mainly changes what you stress.

Croatian does not have a separate present continuous form (like English is being / is doing). The simple present with a time expression covers both:

• On je zauzet ovaj tjedan. – He is (currently) busy this week.
• Ona puno radi ovih dana. – She is working a lot these days.
• Učim hrvatski. – I study Croatian / I am studying Croatian.

The temporary nature is expressed by phrases like ovaj tjedan, ovih dana, trenutno, not by changing the verb form itself.

Yes. Standard Croatian as taught in most courses uses:

• tjedan – week
• vikend – weekend

But regionally (and in Serbian/Bosnian):

• sedmica / sedmica – week (common in Bosnia, parts of Croatia, and in Serbian)
• nedelja / nedjelja – week (Serbian / some dialects)

In Croatia, tjedan and vikend are the standard forms. You may still hear sedmica in some areas, but for learning standard Croatian, it’s safest to stick to tjedan and vikend.