Ako upala ne prođe, morat ću doći opet za tjedan dana.

Ako means if. This is a real/likely conditional sentence: If the inflammation doesn’t go away, I’ll have to come again in a week.

Croatian commonly uses:

• ako + present in the if clause
• a future form in the main clause

So:

• Ako upala ne prođe = If the inflammation doesn’t go away
• morat ću doći... = I’ll have to come...

This is mainly about aspect.

• proći is perfective: it means something reaches completion, here to go away / pass completely
• prolaziti is imperfective: it describes an ongoing process, like to be going away / to pass repeatedly

In this sentence, the speaker means if the inflammation does not go away completely, so Croatian uses the perfective verb proći:

• ne prođe

That is very natural in Croatian when talking about a future result inside an if clause.

If you said ne prolazi, it would sound more like:

• if it isn’t going away
• focusing on the ongoing situation rather than the completed result

Upala is a noun meaning inflammation.

In this sentence it is:

• feminine
• singular
• nominative, because it is the subject of prođe

So:

• upala ne prođe = the inflammation doesn’t go away

In Croatian, ne is usually written as a separate word before a finite verb:

• ne prođe
• ne znam
• ne dolazi

So ne prođe is the normal spelling.

There are a few important exceptions where the negative form is written together, such as:

• neću
• nemam
• nisam

But prođe is not one of those exceptions, so it stays separate.

Morat ću means I will have to.

It is the future tense of morati (to have to / must).

The future in Croatian is formed with:

• the infinitive
• plus a short form of htjeti: ću, ćeš, će, ćemo, ćete, će

With infinitives ending in -ti, when the clitic comes after the infinitive, the final -i is dropped in writing:

• morati + ću → morat ću

So:

• morat ću = I will have to

Yes. Both are possible.

• morat ću doći
• ću morati doći

They mean the same thing: I will have to come.

The version in your sentence, morat ću, is very common and natural.
If the auxiliary comes first, the full infinitive is kept:

• Ja ću morati doći.

If the infinitive comes first, you usually get:

• Morat ću doći.

Because morati is a modal verb, like English must or have to.

Modal verbs are followed by another verb telling you what must be done:

• morati doći = to have to come
• morati čekati = to have to wait
• morati platiti = to have to pay

So:

• morat ću doći = I’ll have to come

Opet means again.

In this sentence:

• doći opet = come again

Its position is somewhat flexible:

• morat ću doći opet za tjedan dana
• morat ću opet doći za tjedan dana

Both are natural. The difference is mostly one of emphasis and rhythm, not basic meaning.

You may also see ponovno, which also means again:

• morat ću ponovno doći

Here za tjedan dana means in a week or a week from now.

This is a very common Croatian time expression:

• za sat vremena = in an hour
• za dva dana = in two days
• za tjedan dana = in a week

So in this sentence it means the person would need to come back one week later.

Tjedan dana is a very common expression meaning a week’s time.

Literally, it is something like a week of days, and dana is the genitive plural of dan (day).

Croatian often uses this kind of structure with periods of time:

• pet dana = five days
• mjesec dana = a month
• godinu dana = a year
• tjedan dana = a week

You can sometimes hear just za tjedan, but za tjedan dana is especially common and idiomatic.

Because za here takes the accusative.

The noun tjedan is masculine inanimate, and for masculine inanimate nouns the accusative singular is the same as the nominative:

• nominative: tjedan
• accusative: tjedan

So:

• za tjedan dana is correct

Tjedna would be genitive singular, which is not what za needs in this expression.

Because Ako upala ne prođe is a subordinate clause placed before the main clause.

Croatian normally separates that kind of opening if clause with a comma:

• Ako upala ne prođe, morat ću doći opet za tjedan dana.

This is similar to English, where a comma is also often used after an opening if clause:

• If the inflammation doesn’t go away, I’ll have to come again in a week.