Την Πρωτοχρονιά θέλω να είμαι με την οικογένειά μου, ακόμα κι αν έχω πολλή δουλειά.

Την Πρωτοχρονιά is in the accusative case, and in Greek the accusative is often used for expressions of time.

So Την Πρωτοχρονιά means on New Year’s Day / at New Year.

A few similar examples:

• Τη Δευτέρα = on Monday
• Τον χειμώνα = in winter
• Την Κυριακή = on Sunday

The article is very normal in Greek with time expressions. Greek uses the article much more often than English does.

Πρωτοχρονιά means New Year’s Day or sometimes New Year in a holiday sense.

It comes from:

• πρώτο- = first
• χρονιά = year

In everyday Greek, Πρωτοχρονιά commonly refers to January 1st, the New Year holiday.

After θέλω (I want), Greek normally uses να + verb.

So:

• θέλω να είμαι = I want to be
• literally: I want that I be

This is a very common Greek pattern:

• Θέλω να πάω = I want to go
• Θέλω να δω = I want to see
• Θέλω να μείνω = I want to stay

English uses the to-infinitive (to be, to go), but Modern Greek does not have an infinitive like English. Instead, it uses να plus a finite verb form.

είμαι is the 1st person singular present form of to be: I am.

In να είμαι, it appears after να, which is the usual way Greek forms clauses like:

• wanting
• possibility
• intention
• purpose

So να είμαι here means to be / that I be, depending on how literally you want to think about it.

The full idea is:

• θέλω να είμαι = I want to be

The preposition με means with, and it is followed by the accusative.

So:

• με την οικογένεια = with the family
• με την οικογένειά μου = with my family

Breakdown:

• με = with
• την = the (feminine accusative singular)
• οικογένεια = family
• μου = my

This is very standard Greek grammar:

• με τον φίλο μου = with my friend
• με τη μητέρα μου = with my mother

In Greek, possessive weak forms like μου (my) usually come after the noun:

• η οικογένειά μου = my family
• το σπίτι μου = my house
• ο φίλος μου = my friend

The accent in οικογένειά μου is a spelling/pronunciation matter. When a word is followed by an enclitic like μου, Greek often adjusts or preserves the accent so the stress pattern is clear.

For a learner, the important thing is to recognize the whole phrase as:

• η οικογένειά μου = my family
• την οικογένειά μου = my family (accusative)

ακόμα κι αν means even if.

It introduces a contrast:

• ακόμα κι αν έχω πολλή δουλειά = even if I have a lot of work

You can think of it as:

• ακόμα = still / even
• κι = and
• αν = if

But as a whole, the expression simply means even if.

Examples:

• Θα πάω, ακόμα κι αν βρέχει. = I’ll go, even if it’s raining.
• Θέλω να έρθεις, ακόμα κι αν αργήσεις. = I want you to come, even if you’re late.

κι is a very common shortened form of και.

So:

• κι = και = and

In expressions like ακόμα κι αν, the shorter κι is especially common and natural.

You may also see the full form ακόμα και αν, which means the same thing.

Because δουλειά is a feminine singular noun, the adjective πολύς (much / many / a lot of) has to agree with it.

Here are the relevant forms:

• πολύς = masculine
• πολλή = feminine
• πολύ = neuter

So:

• πολλή δουλειά = a lot of work
  because δουλειά is feminine

Compare:

• πολλός κόσμος = many people / a lot of people (masculine)
• πολλή ζάχαρη = a lot of sugar (feminine)
• πολύ νερό = a lot of water (neuter)

This is one of the most common things learners ask about, because πολύ is also an adverb meaning very:

• πολύ καλός = very good

But here it is not an adverb. It is an adjective agreeing with δουλειά, so we need πολλή.

The comma separates the main clause from the contrasting subordinate clause.

Main clause:

• Την Πρωτοχρονιά θέλω να είμαι με την οικογένειά μου

Subordinate clause:

• ακόμα κι αν έχω πολλή δουλειά

This is similar to English punctuation in sentences like:

• I want to be with my family, even if I have a lot of work.

Greek punctuation often works in a similar way here.

The word order is quite natural.

Greek is more flexible than English, but this sentence is arranged in a very ordinary way:

• Την Πρωτοχρονιά = time expression first
• θέλω = main verb
• να είμαι = complement of θέλω
• με την οικογένειά μου = with my family
• ακόμα κι αν έχω πολλή δουλειά = even if I have a lot of work

Putting Την Πρωτοχρονιά first gives the sentence a clear time frame right away: As for New Year’s Day...

So the order sounds natural and emphasizes the occasion from the beginning.