Після уроку я ховаю зошит у рюкзак.

Because після requires the genitive case.

• Base form: урок = lesson
• Genitive singular: уроку

So:

• після уроку = after the lesson

This is something you often just have to learn with prepositions: certain prepositions regularly require certain cases.

Ховаю is the 1st person singular present tense form of ховати.

So:

• ховати = to hide / to put away
• я ховаю = I hide / I am hiding / I put away

Ukrainian present tense can often cover both a simple present and a present continuous meaning, depending on context.

It actually is in the accusative case, but for an inanimate masculine noun, the accusative singular often looks exactly like the nominative singular.

So:

• Nominative: зошит
• Accusative: зошит

Since зошит is the direct object of ховаю, it should be in the accusative. It just happens to look the same.

Because here у means into, not in.

With у / в:

• accusative is used for movement/direction into somewhere
• locative is used for location in somewhere

So:

• ховаю зошит у рюкзак = I put/hide the notebook into the backpack
• зошит у рюкзаку = the notebook is in the backpack

That is why рюкзак is accusative here. For this noun, the accusative singular looks like the base form.

Not only literally. In this sentence, ховати can sound like:

• to hide
• to put away
• to stash
• to tuck into

So even if English might naturally say I put my notebook into my backpack after class, Ukrainian can use ховати in a natural way here.

Yes, very often.

Ukrainian commonly drops subject pronouns when the verb already makes the subject clear.

So both are possible:

• Після уроку я ховаю зошит у рюкзак.
• Після уроку ховаю зошит у рюкзак.

The verb ховаю already tells you the subject is I.

Including я may add a little emphasis or just reflect a particular style.

Both у and в can mean in/into. The choice is often made for ease of pronunciation and sound flow.

Ukrainian prefers combinations that are easier to say, so speakers switch between у and в depending on the surrounding sounds.

Here, у рюкзак sounds natural.

In many cases, у and в are interchangeable in meaning, but one may sound smoother than the other in context.

Yes. Ukrainian word order is fairly flexible.

The sentence as given:

• Після уроку я ховаю зошит у рюкзак.

is a natural neutral order: time expression first, then subject, verb, object, destination.

But other orders are also possible, for example:

• Я після уроку ховаю зошит у рюкзак.
• Зошит я ховаю у рюкзак після уроку.

Changing the order usually changes emphasis, not the core meaning.

A useful stress pattern is:

• Пі́сля уро́ку я хова́ю зо́шит у рюкза́к.

Word by word:

• Пі́сля
• уро́ку
• хова́ю
• зо́шит
• рюкза́к

Stress is important in Ukrainian because it is not always predictable and can affect pronunciation clearly.

Because Ukrainian has no articles like English a/an/the.

So зошит can mean:

• a notebook
• the notebook

The exact meaning depends on context.

That is very normal in Ukrainian, and learners need to get used to understanding definiteness from the situation rather than from an article.