Potrzebuję jeszcze jednego wydruku, bo moja koleżanka też ma podobny wniosek.

Because potrzebować usually takes the genitive case in Polish, not the accusative.

So:

• potrzebuję wydruku = I need a printout
• potrzebuję czasu = I need time
• potrzebuję pomocy = I need help

In your sentence, both jednego and wydruku are in the genitive:

• jednego = genitive form of jeden
• wydruku = genitive form of wydruk

So Potrzebuję jeszcze jednego wydruku literally follows the pattern I need one more printout.

For the same reason: after potrzebuję / potrzebować, Polish normally uses the genitive.

Compare:

• dictionary/basic form: jeden wydruk
• after potrzebuję: jednego wydruku

This is a very common thing learners have to get used to in Polish: words often change form depending on what grammar requires.

Here jeszcze means still / yet / another / one more, depending on the context. In this sentence, the most natural English meaning is:

• jeszcze jednego wydruku = one more printout / another printout

So jeszcze adds the idea that the speaker already has some printout(s) or already asked for one, and now needs an additional one.

Wydruk usually means a printout, especially something produced by a printer.

• wydruk = printout, printed copy
• druk can mean print, printing, or printed material in a broader sense

So in this sentence, wydruk is the natural word if someone needs an extra printed copy of a document.

It can mean either, depending on context.

• kolega / koleżanka can mean a friend, classmate, work colleague, or simply someone you know
• it does not always mean a very close friend

In this sentence, moja koleżanka could be:

• my friend
• my female colleague
• my female classmate

The exact meaning depends on the situation.

Też means also / too. In Polish, its position is flexible, but it is usually placed near the word or idea it relates to.

Here:

• moja koleżanka też ma podobny wniosek

This means:

• my friend/colleague also has a similar application

So też tells us that the colleague, like someone else mentioned or implied, has such a document too.

You may hear slightly different word orders, but this one is very natural.

Because here wniosek is the object of ma (has), and with mieć the object is normally in the accusative. Since wniosek is an inanimate masculine noun, its accusative form is the same as its nominative form.

So:

• nominative: podobny wniosek
• accusative: podobny wniosek

That is why the form does not change visibly here.

By contrast, after potrzebuję, you do see the genitive:

• jednego wydruku

So the sentence contains two different case patterns:

• potrzebuję + genitive
• ma + accusative

Wniosek usually means an application, request, or formal submission.

Common examples:

• wniosek o wizę = visa application
• wniosek o urlop = leave request
• złożyć wniosek = to submit an application

In your sentence, podobny wniosek means something like a similar application/formal request.

Yes, ponieważ would also work.

• bo = because
• ponieważ = because / since, a bit more formal or neutral

So:

• Potrzebuję jeszcze jednego wydruku, bo... sounds very natural in everyday speech.
• Potrzebuję jeszcze jednego wydruku, ponieważ... is also correct, but a little more formal.

Not completely. Polish word order is more flexible than English word order, because the endings show grammatical relationships.

The given sentence is natural:

• Potrzebuję jeszcze jednego wydruku, bo moja koleżanka też ma podobny wniosek.

But some variations are possible, for example to change emphasis:

• Bo moja koleżanka też ma podobny wniosek, potrzebuję jeszcze jednego wydruku.
• Potrzebuję jeszcze jednego wydruku, bo też moja koleżanka ma podobny wniosek.
  This is possible, but less natural in many contexts.

So the original version is a good standard sentence to learn.

A rough English-friendly guide is:

• po-tzheh-BOO-yeh

A few helpful points:

• rz sounds like the Polish ż, similar to the s in measure
• ę at the end of -uję is often pronounced very lightly in normal speech, almost like e or nasalized only slightly depending on context
• stress in Polish usually falls on the second-to-last syllable, so here the stress is on bu: potrzeBUję

Usually wydruku is better here.

Why?

• wydruk specifically means a printout
• druk is broader and often refers to print, printing, or printed text in general

If you mean a physical extra copy from a printer, wydruk is the most natural choice.