Сьогодні у мене є багато часу.

A very common way to express possession in Ukrainian is literally something like at me there is:

• у мене = at me / with me
• є = there is / exists

So у мене є багато часу literally looks like At me there is a lot of time, but natural English is I have a lot of time.

This pattern is extremely common in Ukrainian.

Yes. Я маю багато часу is grammatically correct and understandable.

However, у мене є багато часу is often the more natural everyday way to talk about having something available, especially with things like time, money, energy, etc.

A rough comparison:

• У мене є багато часу = I have a lot of time / I’ve got plenty of time
• Я маю багато часу = also I have a lot of time, but it can sound a bit more direct or formal depending on context

Both are good Ukrainian; this sentence uses the very common possession pattern.

Because after у / в in this possession structure, Ukrainian uses the genitive case.

The pronoun я changes like this:

• я = I
• мене = of me / me, in genitive and some other uses
• мені = to me, in dative

So:

• у мене є... = I have... literally at me there is...

You do not say у я.
And у мені would mean something different, more like in me, not I have.

Є is the present-tense existential form meaning is / there is.

In this sentence, it helps build the idea of possession:

• у мене є багато часу = I have a lot of time

In everyday speech, Ukrainians sometimes omit є when the meaning is clear:

• Сьогодні у мене багато часу

That is also natural. Keeping є can make the idea of having / there being a little more explicit.

Because багато requires the following noun to be in the genitive case.

So:

• час = time
• часу = of time / time in the genitive

After quantity words like these, genitive is normal:

• багато часу = a lot of time
• мало часу = little time
• трохи часу = a little time

So багато час is not correct.

Because here час means time as an uncountable idea, not separate countable units.

So Ukrainian uses genitive singular:

• багато часу = a lot of time

If you used часів, that would usually suggest times / periods / eras, which is a different meaning.

So for the ordinary idea of having plenty of time, багато часу is the normal form.

Yes. У мене and в мене both mean the same thing here.

Ukrainian often alternates у and в for smoother pronunciation. This is called euphony. So you may hear both:

• Сьогодні у мене є багато часу
• Сьогодні в мене є багато часу

Both are correct. Very often the choice depends more on sound and rhythm than on grammar.

No. Ukrainian word order is much more flexible than English word order.

Your sentence is perfectly natural:

• Сьогодні у мене є багато часу

But these are also possible:

• У мене сьогодні є багато часу
• Сьогодні в мене багато часу
• Багато часу в мене сьогодні

The difference is usually emphasis, not basic meaning.

For example:

• Сьогодні at the beginning highlights today
• У мене earlier in the sentence can highlight me / as for me
• Багато часу early can emphasize a lot of time

Putting сьогодні first sets the time frame immediately: as for today...

That is very natural in Ukrainian, just as in English you might say:

• Today I have a lot of time

But it can be moved:

• У мене сьогодні є багато часу

That version is also natural. The meaning stays the same; the emphasis shifts slightly.

It can cover all of those ideas depending on the noun.

With час it is best translated as:

• a lot of time
• or simply much time in a more literal gloss

Examples:

• багато людей = many people / a lot of people
• багато води = a lot of water / much water
• багато часу = a lot of time

So in your sentence, багато is the quantity word meaning a lot of.

No. Ukrainian does not have articles like English a/an and the.

So Ukrainian simply says:

• багато часу = a lot of time

The idea that English expresses with articles is usually understood from context in Ukrainian.