Kan du ringa mig så snart som möjligt?

Because Swedish yes/no questions usually put the finite verb first.

• Statement: Du kan ringa mig. = You can call me.
• Question/request: Kan du ringa mig? = Can you call me?

So this is normal Swedish question word order.

It can do either, depending on context, just like English can you.

In this sentence, it is most naturally understood as a request, not a question about physical ability. So Kan du ringa mig ...? is a normal way to ask someone to call you.

If you wanted to sound more explicitly polite or softer, you could also say:

• Skulle du kunna ringa mig ...? = Could you call me ...?

Because kan is a modal verb, and after modal verbs in Swedish, the next verb is normally in the infinitive.

So:

• du ringer = you call / are calling
• du kan ringa = you can call

That is why the sentence uses ringa, not ringer.

After modal verbs such as kan, vill, ska, måste, and får, Swedish normally uses the infinitive without att.

So:

• Jag vill ringa.
• Du kan ringa.
• Hon måste gå.

Not:

• Du kan att ringa ❌

So Kan du ringa mig ...? is exactly what you should expect.

Because mig is the object form of the pronoun, while jag is the subject form.

Compare:

• Jag ringer. = I am calling.
  (jag = subject)
• Ring mig. = Call me.
  (mig = object)

This works like English I/me:

• I call
• Call me

So in Kan du ringa mig ...?, mig is correct because it is the person being called.

Because in standard Swedish, ringa usually takes a direct object.

So you normally say:

• ringa mig
• ringa Anna
• ringa honom

This is similar to English call me.

Learners sometimes expect something like ringa till mig, but in standard Swedish, ringa mig is the normal choice here.

It is a very common fixed expression meaning as soon as possible.

You can think of it as the pattern:

• så + adjective/adverb + som möjligt
• as + adjective/adverb + as possible

Examples:

• så snart som möjligt = as soon as possible
• så fort som möjligt = as fast/soon as possible
• så snabbt som möjligt = as quickly as possible

In your sentence:

• snart = soon
• möjligt = possible

So the whole phrase functions as one time expression.

Yes. Så fort som möjligt is very common and often interchangeable with så snart som möjligt.

Both are natural:

• Kan du ringa mig så snart som möjligt?
• Kan du ringa mig så fort som möjligt?

In many situations, native speakers may use så fort som möjligt even more often in everyday speech, but så snart som möjligt is absolutely correct and natural.

Yes. It is a normal, polite request.

However, because it includes as soon as possible, it can sound a bit urgent. That is not rude by itself, but it depends on context.

If you want to soften it, you could say:

• Kan du ringa mig så snart som möjligt, tack?
• Skulle du kunna ringa mig så snart som möjligt?

The second version sounds a little more formal or gentle.

Because mig is the object, and så snart som möjligt is an adverbial phrase telling when.

In normal Swedish word order, it is very natural to put them like this:

• verb + object + time expression

So:

• ringa mig så snart som möjligt

This is the most straightforward and natural order in this sentence.

In careful writing it is mig, but in everyday speech it is often pronounced more like mej.

So when you listen to spoken Swedish, you may hear:

• Kan du ringa mej ...?

That is normal pronunciation, even though the standard spelling is still mig.