Ona chce zarezerwować stolik w restauracji po przyjeździe do miasta.

Yes. In Polish, subject pronouns are often omitted because the verb form already shows who is doing the action.

So:

• Ona chce zarezerwować stolik... = She wants to book a table...
• Chce zarezerwować stolik... also naturally means She wants to book a table... if the context is clear.

Including ona gives extra emphasis or contrast, for example:

• Ona chce zarezerwować stolik, a on chce zamówić jedzenie na wynos.
  She wants to book a table, and he wants to order takeaway.

Chce is the 3rd person singular form of the verb chcieć = to want.

Present tense of chcieć:

• ja chcę = I want
• ty chcesz = you want
• on/ona/ono chce = he/she/it wants
• my chcemy = we want
• wy chcecie = you want
• oni/one chcą = they want

So ona chce means she wants.

This is about aspect, which is very important in Polish verbs.

• rezerwować = imperfective, focusing on the process or repeated action
• zarezerwować = perfective, focusing on completing the action successfully

After chce (wants), Polish can use either aspect depending on meaning, but here zarezerwować is the natural choice because the idea is she wants to make a reservation successfully, not just be in the process of reserving.

So:

• chce zarezerwować stolik = she wants to book a table
• chce rezerwować stoliki could suggest something more habitual or process-based, like she wants to be booking tables in general

Both words relate to table, but they are used differently.

• stół = table in the general physical sense
• stolik = small table, and very commonly a table at a restaurant/café

In restaurant situations, Polish usually says zarezerwować stolik, not zarezerwować stół.

So stolik is the idiomatic word here.

The preposition w usually means in and requires the locative case when talking about location.

The base form is:

• restauracja = restaurant

In the locative after w, it becomes:

• w restauracji = in the restaurant

This is a standard pattern:

• w domu = in the house
• w pracy = at work
• w szkole = at school
• w restauracji = in the restaurant

It is the locative singular.

The dictionary form is:

• restauracja

After w meaning in, Polish uses the locative:

• w restauracji

This kind of ending change is normal in Polish. Feminine nouns ending in -a often change to -i or -ie in the locative singular.

Po przyjeździe means after arriving or more literally after the arrival.

This is a very common Polish structure:

• po
  • noun in the locative case

Here:

• przyjazd = arrival
• po przyjeździe = after the arrival / after arriving

So instead of using a full clause like after she arrives, Polish often uses a noun phrase like this.

Compare:

• po obiedzie = after lunch
• po pracy = after work
• po przyjeździe = after arriving / after arrival

It comes from the noun przyjazd, which means arrival.

Forms:

• base form: przyjazd
• after po: po przyjeździe

This is the locative singular form.

It comes from the verb:

• przyjechać = to arrive by vehicle / to come by vehicle

So the noun przyjazd is related to the action of arriving.

The preposition do usually means to when talking about movement toward a place, and it requires the genitive case.

Base form:

• miasto = city

After do:

• do miasta = to the city

This is very common:

• do domu = to home / home
• do szkoły = to school
• do Polski = to Poland
• do miasta = to the city

Because the phrase is specifically describing arrival to the city, not arrival in the city or arrival at the restaurant.

• przyjazd do miasta = arrival to the city
• po przyjeździe do miasta = after arriving in/to the city

Polish often uses do with nouns of motion or arrival:

• wyjazd do Polski = trip/departure to Poland
• powrót do domu = return home
• przyjazd do miasta = arrival to the city

So do miasta naturally goes with przyjazd.

Yes, Polish word order is fairly flexible, though some versions sound more natural than others.

Original:

• Ona chce zarezerwować stolik w restauracji po przyjeździe do miasta.

Other possible orders:

• Po przyjeździe do miasta ona chce zarezerwować stolik w restauracji.
• Ona po przyjeździe do miasta chce zarezerwować stolik w restauracji.
• Stolik w restauracji ona chce zarezerwować po przyjeździe do miasta.
  This one is possible, but more marked or emphatic.

The original order is neutral and natural.

It most directly means in a restaurant, but in English the most natural translation is often at a restaurant depending on context.

Polish uses w very broadly for physical location:

• w restauracji
• w szkole
• w pracy

In English, these may become in or at depending on what sounds most natural.

A few words here may be tricky:

• chce ≈ htseh
  The ch sounds like a rough h.
• zarezerwować ≈ za-reh-zer-VO-vatch
• stolik ≈ STO-lik
• restauracji ≈ re-stau-RA-tsyi
• przyjeździe ≈ pshi-YEZH-jye or pshi-YEZ-djye
• miasta ≈ MYA-sta

A few sound notes:

• rz often sounds like zh
• cz sounds like ch in chop
• ć/ci sounds like a soft ch
• ź/zi sounds like a soft zh

Yes, but the structure changes slightly.

• po przyjeździe do miasta = after arriving in the city / after her arrival in the city
• kiedy przyjedzie do miasta = when she arrives in the city

Both can work, but they are not exactly identical in style:

• po przyjeździe do miasta sounds compact and noun-based
• kiedy przyjedzie do miasta uses a full clause and sounds a bit more explicit

Both are natural Polish. The original version is very common and elegant.

Yes. Zarezerwować stolik is the normal, idiomatic way to say book/reserve a table.

Other related expressions:

• zarezerwować miejsce = reserve a seat/place
• dokonać rezerwacji = make a reservation
• mieć rezerwację = have a reservation

So if you are talking about a restaurant table specifically, zarezerwować stolik is exactly what Polish speakers would usually say.