Ben bugün az uyudum.

Yes. In Turkish, the verb ending often already shows the subject, so Ben bugün az uyudum and Bugün az uyudum both mean I slept little today.

• uyudum already means I slept
• so ben is optional

You include ben when you want to:

• add emphasis
• make the subject especially clear
• contrast with someone else

For example:

• Ben bugün az uyudum, ama sen iyi dinlendin.
  I slept little today, but you rested well.

Az means little, not much, or too little, depending on context.

In this sentence, az uyudum means:

• I slept little
• I didn’t sleep much

It describes the amount of sleep.

Compare:

• Çok uyudum = I slept a lot
• Az uyudum = I slept little / not much
• Hiç uyumadım = I didn’t sleep at all

The verb is uyumak = to sleep.

In the past tense:

• remove -mak
• add the past tense marker
• add the personal ending

So:

• uyu- = verb stem used here
• -du = past tense
• -m = I

Together:

• uyu + du + m → uyudum

This is the normal past-tense form of uyumak.

Uyudum = uyu- + -du + -m

• uyu- = sleep
• -du = definite past tense (did, completed action)
• -m = first person singular (I)

So literally it is:

• sleep + past + I
• I slept

This is a very common Turkish pattern:

• geldim = I came
• yaptım = I did
• konuştum = I spoke
• uyudum = I slept

Turkish past tense changes according to vowel harmony and consonant harmony.

The past tense suffix has four vowel forms:

• -dı
• -di
• -du
• -dü

Here, the last vowel before the suffix is u in uyu-, so the suffix becomes -du.

The consonant can also change:

• d after voiced sounds
• t after voiceless sounds

Since the verb stem here ends in a vowel sound, d is used:

• uyu + du → uyudu

Then add -m:

• uyudum

Bugün means today.

It tells you when the action happened:

• Ben bugün az uyudum = I slept little today

Even though today is the present day, the verb is still in the past tense because the sleeping already happened.

English does this too:

• I ate little today
• I worked a lot today
• I slept little today

So bugün does not force the present tense.

Yes, Turkish word order is flexible, although some orders sound more neutral than others.

The most neutral version here is:

• Ben bugün az uyudum.

You could also hear:

• Bugün az uyudum.
• Az uyudum bugün.
• Bugün ben az uyudum.

These all mean roughly the same thing, but the emphasis changes.

For example:

• Bugün az uyudum. → focus on today
• Az uyudum bugün. → focus on sleeping little
• Ben bugün az uyudum. → slightly clearer or more contrastive on I

In Turkish, the word right before the verb often gets special focus.

Yes. In this sentence, az functions adverbially: it modifies the verb uyudum.

It tells us how much the speaker slept.

So:

• az uyudum = I slept little

Because it is not a noun here, it does not need case endings.

You can think of it as similar to English little in:

• I slept little or more naturally:
• I didn’t sleep much

Not exactly.

• az uyudum = I slept little / not much
• biraz uyudum = I slept a little / for a bit

Az often sounds more clearly like insufficient amount.
Biraz often sounds softer and can simply mean a little.

So:

• Bugün az uyudum. → I didn’t get much sleep today.
• Bugün biraz uyudum. → I slept a little today.

They are related, but not identical.

Yes, it is completely natural.

A native speaker might also say:

• Bugün çok az uyudum. = I slept very little today.
• Bugün pek uyumadım. = I didn’t really sleep much today.
• Bu gece az uyudum. = I slept little tonight / last night depending on context

Your original sentence is simple, correct, and natural.

Good question. They are not the same.

• bugün = today
• bu gece = tonight / this night

So:

• Ben bugün az uyudum.
  I slept little today.
  This refers to the day as a whole.

• Ben bu gece az uyudum.
  I slept little tonight / this night.
  This specifically refers to nighttime sleep.

If you are talking about sleep, bu gece can sometimes sound more specific and natural, depending on what you mean.

A simple pronunciation guide is:

• Ben → ben
• bugün → boo-GUEN / more exactly bu-gün
• az → az
• uyudum → oo-yoo-DOOM

A rough full pronunciation:

• Ben bu-gün az oo-yoo-doom

A few notes:

• ü in bugün is a front rounded vowel, which English does not really have
• u in uyudum is like oo
• stress in Turkish is often toward the end, but common time words like bugün can behave a bit differently in natural speech

No, not by itself.

Ben bugün az uyudum specifically means:

• I slept little today
• I didn’t get much sleep today

It does not mean:

• I overslept
• I took a nap
• I slept for a short time on purpose

If you wanted I took a nap, Turkish would usually use something like:

• Bugün biraz uyudum = I slept a little / I napped a bit
  or more explicitly:
• Bugün kestirdim. = I took a nap.

So the original sentence most naturally suggests not enough sleep.