Arkadaşım üzülünce ona çay veriyorum.

Because arkadaşım means my friend.

• arkadaş = friend
• -ım = my

So this is a possessive ending, not a case ending. The subject of the sentence is arkadaşım = my friend.

Üzülünce comes from:

• üzülmek = to get sad, to be upset
• -ınca / -ince / -unca / -ünce = when, once, upon

So:

• üzül-ünce = when he/she gets sad / when my friend gets sad

The suffix changes form because of vowel harmony. Since üzülmek has the vowel ü, the suffix appears here as -ünce.

Because these two verbs are different:

• üzmek = to upset someone
• üzülmek = to get upset / to be sad

In this sentence, the friend is the one experiencing the emotion, so Turkish uses the intransitive verb üzülmek.

Compare:

• Onu üzüyorum. = I am upsetting him/her.
• O üzülüyor. = He/She is getting upset.

That’s right: -ünce does not show person the way a normal finite verb does.

The subject is understood from the noun before it:

• Arkadaşım üzülünce = when my friend gets sad

So even though üzülünce itself does not say he/she/I, the noun arkadaşım tells you who the subject is.

Ona means to him / to her / to it.

It is made from:

• o = he, she, it
• -a / -e = dative ending, often meaning to

So:

• o → ona = to him/her

Here it is the indirect object of vermek (to give), because in Turkish you give something to someone.

You usually know from context, because Turkish does not mark gender in third-person pronouns.

• o can mean he, she, or it
• ona can mean to him, to her, or to it

So the sentence itself does not tell you the friend’s gender.

Because çay here is an indefinite direct object.

In Turkish:

• a specific/definite direct object usually takes the accusative
• a nonspecific/indefinite direct object often stays bare

So:

• çay veriyorum = I give tea / I give some tea / I give a cup of tea
• çayı veriyorum = I give the tea / that specific tea

In your sentence, the tea is not presented as a specific known tea, so çay without accusative is natural.

Veriyorum is the -iyor form, often called the present continuous:

• veriyorum = I am giving

But in Turkish, this form is also often used for present or repeated actions in everyday speech, especially in context.

So here veriyorum can naturally mean:

• I give my friend tea when they get sad

If you said veririm, that would sound more like a general habit, rule, or characteristic:

• Arkadaşım üzülünce ona çay veririm. = When my friend gets sad, I give them tea.

Both are possible, but veriyorum is very common in natural speech.

A very literal breakdown is:

• Arkadaşım = my friend
• üzülünce = when (he/she) gets sad
• ona = to him/her
• çay = tea
• veriyorum = I am giving / I give

So the structure is basically:

[When my friend gets sad] [to him/her] [tea] [I give].

This is normal Turkish word order:

• the verb usually comes at the end
• time/condition clauses often come before the main clause

Usually üzülünce most directly suggests when my friend gets upset / becomes sad.

If you want a more clearly stative idea, Turkish often uses something like:

• Arkadaşım üzgün olunca = when my friend is sad

So there is a small difference:

• üzülünce = when they get upset / become sad
• üzgün olunca = when they are sad

In many English translations, though, both may simply come out as when my friend is sad.

Yes, if the context is clear.

For example:

• Üzülünce ona çay veriyorum. = When he/she gets sad, I give him/her tea.
• Arkadaşım üzülünce çay veriyorum. = When my friend gets sad, I give tea.

Turkish often drops things that are already understood from context. But in the full sentence, arkadaşım and ona make everything clear:

• who gets sad
• who receives the tea