Po pracy kupuję arbuza, ananasa i dwie brzoskwinie.

Because po meaning after takes the locative case here. The base noun is praca (work), and its locative singular form is pracy.

So:

• praca = work
• po pracy = after work

This is a very common phrase in Polish.

Polish often drops subject pronouns because the verb ending already tells you who is doing the action.

• kupuję = I buy / I am buying

So ja is not needed. You can add ja only for emphasis or contrast:

• Ja kupuję, a on gotuje. = I’m buying, and he’s cooking.

Polish does not have articles. There is no separate word for a, an, or the.

So:

• kupuję arbuza can mean I’m buying a watermelon
• and in the right context it could also mean I’m buying the watermelon

The exact meaning comes from context, not from an article.

Kupuję is the 1st person singular present tense of the imperfective verb kupować. It can mean:

• I buy
• I am buying

In this sentence, it often sounds like a routine or repeated action: After work, I buy...

Kupię is different. It is the perfective future, meaning:

• I will buy
• I’ll buy

So:

• Po pracy kupuję... = a habit / general present
• Po pracy kupię... = a future, one-time completed action

Because they are the direct objects of kupuję, so they are in the accusative case.

For these nouns, the accusative singular is:

• arbuz → arbuza
• ananas → ananasa

A useful thing to know is that many masculine inanimate nouns keep the same form in nominative and accusative, but some common nouns—especially many food words—take -a in the accusative singular. These two nouns belong to that group.

Because brzoskwinia is a feminine noun. In Polish, two changes depending on gender.

• dwa is used with many masculine and neuter nouns
• dwie is used with feminine nouns

So:

• dwa jabłka = two apples
• dwie brzoskwinie = two peaches

Because the sentence means two peaches, so the noun must be plural.

• brzoskwinię = a peach (singular accusative)
• brzoskwinie = peaches (plural)

After dwie, you use the plural form, so dwie brzoskwinie is correct.

Yes. Polish word order is fairly flexible.

The original:

• Po pracy kupuję arbuza, ananasa i dwie brzoskwinie.

is very natural and puts po pracy first, which highlights the time phrase after work.

You could also say:

• Kupuję po pracy arbuza, ananasa i dwie brzoskwinie.

That is also possible, but the emphasis changes slightly. The original sentence sounds very natural and straightforward.