Hoje eu tenho muito trabalho.

Yes. You can say both:

• Hoje eu tenho muito trabalho.
• Hoje tenho muito trabalho.

In Brazilian Portuguese, subject pronouns like eu are often optional because the verb form already shows who the subject is. Tenho clearly means I have.

Including eu can make the sentence sound a little clearer, more explicit, or slightly more emphatic, but it is not required here.

Tenho is the first-person singular form of the verb ter, which means to have.

Present tense of ter:

• eu tenho = I have
• você tem = you have
• ele/ela tem = he/she has
• nós temos = we have
• vocês têm = you all have
• eles/elas têm = they have

So in this sentence, tenho means I have.

Here, trabalho means work in the general, uncountable sense, like a lot of work in English.

So:

• muito trabalho = a lot of work

If you say muitos trabalhos, that usually means many works / many assignments / many papers / many jobs, depending on context. That is a more countable meaning.

So for the idea of being busy, muito trabalho is the natural choice.

Here it is a noun.

Why?

Because after tenho (I have), the word trabalho is the thing being had:

• tenho trabalho = I have work

The word trabalho can also be a verb form:

• eu trabalho = I work

So trabalho can be either:

• noun: work, job
• verb: I work

In this sentence, it is definitely a noun.

Because trabalho is a masculine singular noun.

The word muito must agree with the noun when it works like a lot of / much before a noun:

• muito trabalho = a lot of work
• muita água = a lot of water
• muitos livros = many books
• muitas coisas = many things

Since trabalho is masculine singular, muito is the correct form.

Portuguese often does not use an article in expressions like this, especially with uncountable or general nouns after ter.

So:

• tenho muito trabalho = I have a lot of work

This is natural and idiomatic.

You are talking about work in general, not a specific piece of work. That is why no article is needed.

Yes. Portuguese word order is fairly flexible.

All of these are possible:

• Hoje eu tenho muito trabalho.
• Eu tenho muito trabalho hoje.
• Tenho muito trabalho hoje.

The meaning stays basically the same, but the emphasis changes a little:

• Hoje eu tenho muito trabalho puts today up front.
• Eu tenho muito trabalho hoje sounds a bit more neutral in some contexts.
• Tenho muito trabalho hoje is also very natural in conversation.

In standard Brazilian Portuguese, hoje is roughly pronounced like:

• OH-zhee or OH-zheh

A more careful approximation is HO-zhi, where the middle sound is like the s in measure or the g in genre.

A few pronunciation notes:

• h is silent
• j in Portuguese usually has the zh sound
• the final e is often pronounced like a light ee sound in many Brazilian accents

So hoje does not sound like English hodge or ho-gee.

It is natural, but there are other very common ways to express the same idea in Brazilian Portuguese.

For example:

• Hoje tenho muito trabalho.
• Hoje estou com muito trabalho.
• Hoje vou trabalhar muito. = Today I’m going to work a lot.
• Hoje estou muito ocupado/ocupada. = Today I’m very busy.

Small differences:

• tenho muito trabalho focuses on having a lot of work
• estou com muito trabalho is also very common and conversational
• estou ocupado/ocupada focuses more on being busy

Here it means work, not job.

In Portuguese, trabalho can mean different things depending on context:

• work in general
• job
• assignment / paper / project

In Hoje eu tenho muito trabalho, the intended meaning is I have a lot of work to do today.

If you wanted to say I have a job, that would usually need a different context, for example:

• Eu tenho um trabalho.
  But in Brazilian Portuguese, Eu tenho um emprego is often clearer for I have a job.

Usually, no comma is needed:

• Hoje eu tenho muito trabalho.

That is the normal way to write it.

You might see a comma if the writer wants to mark a pause or add emphasis:

• Hoje, eu tenho muito trabalho.

But in ordinary writing, the version without the comma is more standard.