أنتظر ساعة تقريبا.

Because the verb already tells you the subject.

أنتظر means I wait / I am waiting / I will wait, and the أـ at the beginning marks first person singular. So Arabic often leaves out the separate pronoun أنا unless it is needed for emphasis or contrast.

• أنتظر ساعةً تقريبًا = I’m waiting about an hour
• أنا أنتظر ساعةً تقريبًا = I am waiting about an hour (with extra emphasis on I)

أنتظر is the imperfect (non-past) form of the verb انتظر.

In Modern Standard Arabic, the imperfect can express:

• present: I wait / I am waiting
• sometimes future, depending on context: I will wait

So أنتظر does not only mean one tense in the same way English does. Context decides.

Its dictionary form is:

• انتظر = to wait

In Arabic, expressions of duration often do not need a preposition like English for.

So:

• ساعةً literally = an hour
• but in this sentence it means for an hour

This is a common Arabic structure: a time word can function as an accusative of duration.

So:

• أنتظر ساعةً تقريبًا = I wait/am waiting for about an hour

English needs for, but Arabic often does not.

Because everyday Arabic writing usually leaves out short vowels and case endings.

The fully vocalized form would be:

أنتظرُ ساعةً تقريبًا

Here:

• أنتظرُ ends with ـُ
• ساعةً ends with ـً because it is an indefinite accusative noun
• تقريبًا also has tanween fatḥ

So the sentence as written is normal unvocalized Arabic, but the grammatical ending is still understood.

تقريبًا means approximately, roughly, or about.

It modifies the time expression ساعة, so:

• ساعة تقريبًا = about an hour

This is a very common way to soften or approximate a quantity in Arabic.

Other common options are:

• حوالي ساعة = around an hour
• نحو ساعة = about an hour

Yes. The Arabic imperfect often covers both ideas.

So depending on context, أنتظر ساعةً تقريبًا could be understood as:

• I am waiting about an hour
• I wait about an hour
• I will wait about an hour

If you want to make the future especially clear, Arabic often adds:

• سأنتظر ساعةً تقريبًا = I will wait about an hour

Possibly, but not necessarily. Arabic often leaves out information that is already understood from context.

If the object is stated, انتظر usually takes it directly, without a preposition like English for:

• أنتظر الحافلة = I am waiting for the bus
• أنتظر صديقي = I am waiting for my friend

So if you wanted to include both the thing/person and the duration, you could say:

• أنتظر الحافلة ساعةً تقريبًا = I am waiting for the bus for about an hour

In your sentence, the object is simply not mentioned.

A fully vocalized version is:

أنتظرُ ساعةً تقريبًا

A careful pronunciation would be roughly:

antaẓiru sāʿatan taqrīban

A few pronunciation notes:

• ظ in أنتظر is a heavy/emphatic sound
• ع in ساعة is a distinct Arabic consonant
• the final ة in ساعةً is pronounced with the case ending here, so not just a plain -ah in fully read MSA

Yes. A few common alternatives are:

• أنتظر حوالي ساعة = I wait/am waiting about an hour
• أنتظر نحو ساعة = I wait/am waiting about an hour
• أنتظر لمدة ساعة تقريبًا = I wait/am waiting for about an hour

The original sentence is perfectly understandable, and ساعةً تقريبًا is a very common pattern. لمدة just makes the idea of duration more explicit.