W tym roku na Sylwestra zaprosimy tylko rodzinę i przyjaciół.

Because the preposition w meaning in takes the locative case here.

• ten rok = this year in the dictionary/basic form
• after w, it changes to tym roku

So:

• ten → tym
• rok → roku

This is a very common pattern:

• w tym domu = in this house
• w tym mieście = in this city
• w tym roku = in this year / this year

In Polish, w tym roku is the normal idiomatic way to say this year in the sense of during this year.

Both can exist, but they are used a little differently.

In this sentence, na Sylwestra means something like for New Year’s Eve or for the New Year’s Eve celebration. It fits well with invite because you invite someone to/for an occasion.

• zaprosić kogoś na imprezę = invite someone to a party
• zaprosić kogoś na obiad = invite someone for dinner
• zaprosić kogoś na Sylwestra = invite someone for New Year’s Eve

By contrast, w Sylwestra usually means on New Year’s Eve, focusing more on the time/date:

• W Sylwestra zostajemy w domu. = On New Year’s Eve, we’re staying home.

So in this sentence, na Sylwestra is the natural choice because it is about the event people are being invited to.

Because after na in this meaning, Polish uses the accusative case.

So:

• Sylwester = basic form
• na Sylwestra = for/on New Year’s Eve

This is the same kind of change you see in many masculine nouns after na:

• na obiad = for dinner
• na koncert = to a concert
• na Sylwestra = for New Year’s Eve

So Sylwestra is not a different word; it is the case form required by the grammar.

Because zaprosimy comes from the perfective verb zaprosić, which is used for a completed, one-time action.

Here the idea is:

• we will invite them
• as a single planned action
• and the invitation will be completed

That is why zaprosimy is the most natural form.

Compare:

• zaprosimy = we will invite
• będziemy zapraszać = we will be inviting / we will invite repeatedly or as a process

In this sentence, the speaker is talking about one future decision or act, so zaprosimy fits better.

Because Polish verbs usually show the subject in the ending.

The ending -my in zaprosimy tells you the subject is we.

So:

• zaproszę = I will invite
• zaprosisz = you will invite
• zaprosi = he/she/it will invite
• zaprosimy = we will invite
• zaprosicie = you all will invite
• zaproszą = they will invite

That is why Polish often leaves out subject pronouns like ja, ty, my, unless they are needed for emphasis or contrast.

Because Polish does not have articles like English a/an/the.

So Polish does not need separate words for:

• the family
• the friends
• a family
• some friends

Instead, Polish usually shows definiteness from context.

In this sentence, English might naturally say the family and friends or just family and friends, but Polish simply says:

• rodzinę i przyjaciół

The listener understands from context whose family and friends are meant.

Because rodzinę is the accusative singular form, and it is the direct object of zaprosimy.

The basic form is:

• rodzina = family

But after a transitive verb like zaprosić when family is the thing/person being invited, Polish uses the accusative:

• zaprosimy rodzinę = we will invite the family

For many feminine nouns ending in -a, the accusative singular ends in -ę:

• kobieta → kobietę
• siostra → siostrę
• rodzina → rodzinę

Because przyjaciół is the form needed for the accusative plural of a masculine personal noun.

The basic plural is:

• przyjaciele = friends

But after zaprosimy, the noun is a direct object, so it must change case. For masculine personal nouns, the accusative plural usually looks like the genitive plural:

• przyjaciele → przyjaciół

So:

• przyjaciele przyszli = the friends came
• zaprosimy przyjaciół = we will invite friends / the friends

This is a very important pattern in Polish.

Yes, rodzinę is grammatically singular.

That is completely normal, because rodzina in Polish, like family in English, is a collective noun: one word referring to a group of related people.

So zaprosimy rodzinę means we will invite the family as a group.

If you wanted families in the plural, you would need a different form, such as rodziny, depending on the case and sentence structure.

In this sentence, tylko means only, and it most naturally limits rodzinę i przyjaciół.

So the idea is:

• we will invite only family and friends
• not coworkers, neighbors, acquaintances, etc.

Because tylko is placed right before rodzinę i przyjaciół, that is what it focuses on.

If you moved tylko, the emphasis could shift slightly. For example:

• Tylko na Sylwestra zaprosimy rodzinę i przyjaciół.
  = It is only for New Year’s Eve that we’ll invite family and friends.

• Na Sylwestra zaprosimy tylko rodzinę i przyjaciół.
  = For New Year’s Eve, we’ll invite only family and friends.

So the sentence as given most strongly emphasizes who will be invited.

Yes. Polish word order is much more flexible than English word order, because the case endings show grammatical roles.

The given sentence is natural:

• W tym roku na Sylwestra zaprosimy tylko rodzinę i przyjaciół.

But you could also say:

• Na Sylwestra w tym roku zaprosimy tylko rodzinę i przyjaciół.
• Zaprosimy w tym roku na Sylwestra tylko rodzinę i przyjaciół.
• W tym roku zaprosimy na Sylwestra tylko rodzinę i przyjaciół.

These versions all mean roughly the same thing, but the focus changes a bit:

• what is new information
• what is being contrasted
• what the speaker wants to emphasize first

The original order is very natural because it starts with the time frame w tym roku, then the occasion na Sylwestra, then the action zaprosimy, and finally the restricted group tylko rodzinę i przyjaciół.