Ohne Leiter komme ich nicht an den Koffer auf dem Schrank.

German main clauses follow the verb-second rule. That means the finite verb must be in the second position.

Here, Ohne Leiter is placed first for emphasis, so the verb komme has to come next, and the subject ich comes after it:

Ohne Leiter komme ich nicht an den Koffer auf dem Schrank.

If you started with the subject instead, that would also be correct:

Ich komme ohne Leiter nicht an den Koffer auf dem Schrank.

The version with Ohne Leiter first puts extra focus on the fact that there is no ladder.

Both are possible, but they do not feel exactly the same.

• ohne Leiter = without a ladder / without any ladder, in a general sense
• ohne eine Leiter = without a ladder, but with slightly more emphasis on one particular ladder or on the countable object

German often leaves out the article when talking about tools, materials, or things in a general way. So ohne Leiter sounds very natural here.

Also, ohne takes the accusative case. If you do use the article, it would be:

• die Leiter → nominative
• eine Leiter → accusative after ohne

Because the two prepositions work differently here.

• an den Koffer: here an is part of the expression an etwas kommen, meaning to get to / to reach something. It uses the accusative here, so der Koffer becomes den Koffer.
• auf dem Schrank: auf is a two-way preposition. Since this is a location and not a movement onto something, German uses the dative. So der Schrank becomes dem Schrank.

Compare:

• Der Koffer liegt auf dem Schrank. = The suitcase is on the cupboard.
• Ich stelle den Koffer auf den Schrank. = I put the suitcase onto the cupboard.

So dem Schrank is dative because the suitcase is already there.

Here an den Koffer kommen means to get to the suitcase or to reach the suitcase.

It does not mean to arrive at the suitcase in the usual come sense. This is a common German way of expressing access or reach.

So:

• Ich komme an den Koffer. = I can reach the suitcase.
• Ich komme nicht an den Koffer. = I can’t reach the suitcase.

Many speakers also say:

• an den Koffer herankommen
• an den Koffer rankommen

Those versions make the meaning reach / get at even clearer.

In German, nicht usually comes before the part of the sentence that it negates.

Here, the idea being negated is an den Koffer auf dem Schrank kommen — reaching the suitcase on top of the cupboard. So nicht appears before that part:

Ohne Leiter komme ich nicht an den Koffer auf dem Schrank.

That is the natural placement.

If you rearrange the sentence, nicht will often stay near the part being negated:

Ich komme ohne Leiter nicht an den Koffer auf dem Schrank.

So the position of nicht is connected to meaning, not just fixed by one simple rule.

It means on top of the cupboard / wardrobe / cabinet.

It does not mean inside the cupboard.

That distinction is important:

• auf dem Schrank = on the cupboard / on top of it
• im Schrank = in the cupboard / inside it

So the suitcase is sitting on top of the Schrank, not inside it.

Schrank is a fairly broad word. Depending on context, it can mean:

• cupboard
• cabinet
• wardrobe

In this sentence, cupboard or cabinet is probably the best fit, because the suitcase is on top of it.

German furniture words do not always match one exact English word, so the best translation depends on the situation.

Yes. That is also correct and natural.

The difference is mainly emphasis:

• Ohne Leiter komme ich nicht an den Koffer auf dem Schrank.
  Focuses more on without a ladder
• Ich komme ohne Leiter nicht an den Koffer auf dem Schrank.
  Starts more neutrally with I

German lets you move things around fairly freely in main clauses, as long as the finite verb stays in second position.

Yes, it can be.

German has:

• die Leiter = ladder
• der Leiter = leader / manager / head of something

They are spelled the same, but the gender changes the meaning.

In your sentence, the meaning is clearly ladder, because the context is reaching a suitcase on top of a cupboard. A leader would not make sense there.

If an article were present, it would be easier to see immediately:

• ohne eine Leiter = without a ladder
• ohne einen Leiter = without a male leader/manager

Yes. The given sentence is understandable and natural, but there are other common ways to say the same thing.

For example:

• Ohne Leiter kann ich den Koffer auf dem Schrank nicht erreichen.
• Ohne Leiter komme ich nicht an den Koffer auf dem Schrank heran.
• Ohne Leiter komme ich nicht an den Koffer auf dem Schrank ran. (more colloquial)

These all express the same basic idea.
The version with erreichen is a bit more direct and easy for learners, while an etwas herankommen / rankommen is very common in everyday German.