هذا شخص عجوز.

That is normal in Arabic. In a present-tense nominal sentence, Arabic usually does not use a separate word for is/are.

So:

• هذا شخص عجوز
• literally looks like this person old
• but naturally means This is an old person

If you want the past or future, Arabic does use verbs, such as كان for was.

هذا means this and is the masculine singular demonstrative.

It fits naturally here because شخص is grammatically masculine in Arabic, even though its meaning is general and can refer to a person of either sex.

Also, هذا can stand by itself at the start of a sentence to mean this is ...

Compare:

• هذا شخص عجوز = This is an old person
• هذا الشخص = this person

Because the sentence means This is an old person, not This person is old.

In Arabic, an indefinite noun after هذا often gives the meaning this is a ...

Compare the two patterns:

• هذا شخص عجوز = This is an old person
• هذا الشخص عجوز = This person is old

So adding الـ changes the structure and the meaning.

Because Arabic adjectives normally come after the noun they describe.

So:

• شخص عجوز = an old person
• not عجوز شخص

The adjective also agrees with the noun in things like definiteness. Here both words are indefinite:

• شخص = an व्यक्ति/person
• عجوز = old

If the noun were definite, the adjective would also be definite:

• الشخص العجوز = the old person

Yes. عجوز is commonly used with both masculine and feminine singular nouns without changing form.

For example:

• رجل عجوز = an old man
• امرأة عجوز = an old woman

So if you wanted a feminine version of the sentence, a natural example would be:

• هذه امرأة عجوز = This is an old woman

Because عجوز is used for a person who is old in age, especially elderly.

قديم usually means old for things, in the sense of old, ancient, former, long-standing.

So:

• شخص عجوز = an elderly person
• كتاب قديم = an old book
• بيت قديم = an old house

Using قديم for a person's age would sound wrong in standard usage.

A careful full pronunciation is:

• hādhā shakhṣun ʿajūzun

In normal everyday reading, where case endings are often not pronounced, you may hear:

• hādhā shakhṣ ʿajūz

A few sound notes:

• ذ in هذا sounds like th in this
• خ in شخص is a rough kh sound, like in Scottish loch or German Bach
• ع in عجوز is a deep throat sound with no exact English equivalent

Fully vocalized, the sentence is:

• هٰذا شَخْصٌ عَجوزٌ

The endings -ٌ on شخص and عجوز are the indefinite nominative endings, often written as -un in transliteration:

• shakhṣun
• ʿajūzun

In most normal Arabic writing, these short vowels and case endings are omitted, so you usually just see:

• هذا شخص عجوز

That does not mean the grammar is different; the endings are just unwritten.