Potrzeba mi więcej czasu.

Because this sentence uses a different pattern from the ordinary verb potrzebować.

• Potrzebuję więcej czasu = the most standard, everyday way to say I need more time.
• Potrzeba mi więcej czasu = a more impersonal, slightly more literary or elevated structure.

In potrzeba mi więcej czasu, potrzeba is not a normal 1st-person verb form. It is part of a fixed expression meaning roughly there is a need, for me, of more time.

So both sentences can mean the same thing, but Potrzebuję więcej czasu is more neutral in everyday speech.

Because the person who feels the need is put in the dative case in this construction.

So:

• mi = to me / for me
• nam = to us / for us
• mu = to him / for him
• jej = to her / for her

This pattern is:

potrzeba + dative person + genitive thing

Examples:

• Potrzeba mi spokoju. = I need peace.
• Potrzeba nam pomocy. = We need help.

Both forms are related, but mi is the usual unstressed short form of the dative pronoun.

• mi = normal in neutral sentences
• mnie = used for emphasis, contrast, or in some positions where the full form sounds better

So:

• Potrzeba mi więcej czasu. = normal
• Mnie potrzeba więcej czasu. = more emphatic, like I’m the one who needs more time

In most ordinary cases here, mi is the expected form.

Because this construction requires the thing needed to be in the genitive case.

So:

• czas = nominative
• czasu = genitive

With potrzeba komuś czegoś, you normally use the genitive:

• Potrzeba mi czasu.
• Potrzeba jej odpoczynku.
• Potrzeba nam pieniędzy.

So czasu is not accidental; it is required by the grammar of the expression.

Yes. In fact, czasu is especially natural here for two reasons:

1. potrzeba takes the genitive
2. więcej also normally takes a noun in the genitive

So:

• więcej czasu = more time
• więcej pieniędzy = more money
• więcej pracy = more work

That means czasu is exactly what you would expect after więcej.

No, not in normal usage.

You do not need to say:

• Jest potrzeba mi więcej czasu — this is not correct here.

Potrzeba mi więcej czasu is already a complete sentence.

Polish often allows compact, subjectless, impersonal expressions, and this is one of them.

It is natural, but it is less neutral and everyday than Potrzebuję więcej czasu.

It can sound:

• a bit literary
• a bit elevated
• sometimes slightly old-fashioned
• sometimes more expressive

A native speaker would absolutely understand it, but in ordinary conversation many people would more readily say:

Potrzebuję więcej czasu.

So the sentence is correct and natural, just stylistically marked.

Yes. Polish word order is flexible because the case endings show the grammatical relationships.

For example, you may also hear:

• Więcej czasu mi potrzeba.
• Mi potrzeba więcej czasu.

These versions are still understandable, but the emphasis changes.

• Potrzeba mi więcej czasu. = fairly neutral
• Więcej czasu mi potrzeba. = stronger focus on more time

So the original sentence is a normal word order, but not the only possible one.

A rough literal breakdown is:

• potrzeba = need
• mi = to me / for me
• więcej czasu = more time

So very literally, it is something like:

There is a need for me of more time.

That is not natural English, but it helps explain why the Polish grammar looks different from I need more time.

Yes. The same structure works with many nouns.

Examples:

• Potrzeba mi spokoju. = I need peace.
• Potrzeba nam odpoczynku. = We need rest.
• Potrzeba jej pomocy. = She needs help.
• Potrzeba im pieniędzy. = They need money.

So the pattern is productive, even if it is more stylistically marked than the ordinary verb potrzebować.

The main thing to remember is this:

• Potrzebuję więcej czasu = the most common everyday way
• Potrzeba mi więcej czasu = correct, natural, but more impersonal/literary

And grammatically:

• the person is in the dative: mi
• the thing needed is in the genitive: czasu

So if you understand dative person + genitive thing, the sentence becomes much easier to read.