Nevím, kterým směrem mám jít na poštu.

Czech often drops subject pronouns because the verb ending already shows the person.

• nevím = I don't know
• mám = I have / I am supposed to

So já is not necessary here. You could say Já nevím..., but that usually adds emphasis, like As for me, I don't know...

Kterým is a form of který = which.

Here it agrees with směrem:

• směr = direction
• směrem = instrumental singular of směr
• therefore kterým must also be instrumental singular masculine

So kterým směrem means in which direction / which way.

This is a fixed and very common Czech expression:

• kterým směrem = which direction / which way
• tím směrem = that way / in that direction

The noun směr becomes směrem in the instrumental when used in this kind of adverbial phrase. English does not mark this with case, but Czech does.

Yes, you often can, but there is a small nuance.

• kterým směrem = which direction, often when choosing among possible ways
• jakým směrem = in what direction / in what way

In everyday speech, both can work in this sentence. Kterým směrem sounds very natural for asking or wondering which way to go.

Literally, mám jít is I am to go / I am supposed to go / I should go.

In this sentence, mám adds the idea of what I should do or what way I am supposed to take. So the sentence is not just about movement, but about uncertainty over the correct choice.

Compare:

• Nevím, kterým směrem jít na poštu. = I don't know which way to go to the post office.
• Nevím, kterým směrem mám jít na poštu. = I don't know which way I should go to the post office.

The version with mám can sound slightly more like what am I supposed to do?

Yes. That is a perfectly natural sentence.

Without mám, it sounds a bit more direct and neutral:

• Nevím, kterým směrem jít na poštu.

With mám, there is a slight extra nuance of which way should I go / am I supposed to go:

• Nevím, kterým směrem mám jít na poštu.

Both are correct.

Because Czech idiomatically says jít na poštu for go to the post office.

This is just how certain destinations are expressed in Czech. With some places, Czech uses na where English uses to and where a learner might expect do.

Here:

• na poštu = to the post office

Do pošty is generally not the normal way to say this when you mean going to the post office as a destination.

Because na takes the accusative when it expresses movement toward a destination.

So:

• jdu na poštu = I am going to the post office
  (na
  • accusative)

Compare with location:

• jsem na poště = I am at the post office
  (na
  • locative)

This is a very important Czech pattern:

• motion toward a place -> often accusative
• location at a place -> locative

It can mean both, depending on context.

• pošta = mail / post
• pošta = post office

In this sentence, because of jít na poštu (go to the post office), it clearly means post office.

Jít usually means to go on foot.
Jet usually means to go by vehicle.

So:

• jít na poštu = go to the post office, probably walking
• jet na poštu = go to the post office by car, bus, etc.

If you are talking generally about how to get there, jít is very common, especially for short distances in town.

The order here is very natural and neutral:

• Nevím, kterým směrem mám jít na poštu.

The first part, Nevím, is the main clause: I don't know.
The rest is an indirect question: which way I should go to the post office.

Czech word order is fairly flexible, but changing it changes emphasis. For example:

• Nevím, kterým směrem mám jít na poštu. = neutral
• Nevím, na poštu kterým směrem mám jít. = possible, but marked
• Nevím, kterým směrem na poštu mám jít. = also possible, with slightly different focus

For learners, the original order is the safest and most natural choice.

It contains an indirect question, not a direct one.

• Direct question: Kterým směrem mám jít na poštu? = Which way should I go to the post office?
• Indirect question: Nevím, kterým směrem mám jít na poštu. = I don't know which way I should go to the post office.

Because the whole sentence is a statement (I don't know...), it ends with a period, not a question mark.