انا مش هروح السوق النهارده عشان عندي مشوار.

A natural pronunciation is:

ana mish haruuH es-suu' ennaharda 'ashan 'andi mashwar

A few notes:

• مش = mish
• هروح = haruuH
  The last sound is the Arabic ح, a stronger, breathier h than in English.
• السوق = es-suu' or is-suu'
• النهارده = ennaharda
• عشان = 'ashan
• عندي = 'andi
• مشوار = mashwar or mashwaar

You will often hear the whole sentence said pretty smoothly, without strong pauses.

مش is the negative particle here. It makes the sentence negative:

• هروح السوق = I will go to the market
• مش هروح السوق = I will not go to the market / I am not going to the market

In Egyptian Arabic, مش is very common for negating nominal sentences and also future expressions like this one.

The هـ at the beginning is the Egyptian future marker.

• أروح = I go / I am going / I go regularly, depending on context
• هروح = I will go / I’m going to go

So:

• أنا هروح السوق = I’ll go to the market
• أنا مش هروح السوق = I won’t go to the market

This is one of the most common ways to talk about the future in Egyptian Arabic.

That is the normal Egyptian pattern for this kind of future sentence:

• مش + future verb

So:

• مش هروح
• مش هاشتغل
• مش هاكل

It would sound unnatural to move مش after the verb. The usual order is exactly what you see here.

You can usually leave it out.

Because هروح already shows first person singular, the subject is understood. So both are natural:

• أنا مش هروح السوق النهارده عشان عندي مشوار
• مش هروح السوق النهارده عشان عندي مشوار

Including أنا can add a little emphasis, clarity, or contrast, especially if you are comparing yourself with someone else.

With راح / يروح / أروح in Egyptian Arabic, destinations are often used directly, without a separate word for to.

So:

• أروح السوق = go to the market
• أروح الشغل = go to work
• أروح البيت = go home

This is very normal in Egyptian Arabic. English needs to, but Egyptian Arabic often does not in this pattern.

Also, السوق literally means the market, not just market.

النهارده means today in Egyptian Arabic.

It is a very common colloquial Egyptian word. In Modern Standard Arabic, you would more commonly see اليوم for today.

Historically, النهارده comes from something like this day, and in speech it behaves like one fixed everyday word meaning today.

Here, عشان means because.

So this part:

• عشان عندي مشوار = because I have an errand / because I’ve got something to do

A useful thing to know is that عشان can also mean so that / in order to in other contexts. The meaning depends on the sentence.

Here it is clearly giving a reason, so because is the best translation.

This is a very common Arabic structure.

• عند basically means something like at or with
• عندي literally means at me or with me

But in natural English, it is usually translated as I have.

Examples:

• عندي عربية = I have a car
• عندي وقت = I have time
• عندي مشوار = I have an errand / I’ve got somewhere to go / I have something to do

So Arabic often expresses possession in a different way from English.

مشوار is a very useful everyday Egyptian word. Its meaning is broad.

It can mean:

• an errand
• a trip
• an outing
• something you need to go do
• a place you need to go for some purpose

So عندي مشوار does not only mean one specific errand like buying milk. It often means something more general, like:

• I have somewhere to go
• I have something to take care of
• I’ve got an errand to run

That is why this phrase is so common in conversation.

Yes. The sentence you were given is very natural and neutral:

• أنا مش هروح السوق النهارده عشان عندي مشوار

But Egyptian Arabic allows some flexibility, depending on what you want to emphasize.

For example:

• النهارده مش هروح السوق عشان عندي مشوار
  This emphasizes today.
• عشان عندي مشوار، مش هروح السوق النهارده
  This puts the reason first.
• مش هروح السوق النهارده، عندي مشوار
  This sounds a bit more conversational and broken into two parts.

So the basic meaning stays the same, but the emphasis can shift.