In the early days when laptops became popular someone invented the word läppäri (← laptop) and it has stayed in the colloquial use, but in normal speech one would say today Eikö sinulla ole kannettavaa (Don't you have a portable?) or just Eikö sinulla ole tietokonetta mukanasi? (Don't you have a computer with you?).
Let's look at a positive statement first. In the English sentence "You have a laptop", "you" is the subject and "a laptop" the object. The case system in English has been greatly reduced during the centuries up to this point where the object form or the accusative of "a laptop" is the same as its nominative form. (Cf. with another germanic language German.)
Finnish expresses possesion other way, by "Sinulla on läppäri", which is actually a special case of so called existential clauses ("something is somewhere"; here "a laptop is by you" = "you have a laptop"). So the subject is läppäri and sinulla is an adverbial describing the location.
How about a negative statement? In English you just add the negative particle "You do not have a laptop". If you negate a possessive statement in Finnish, the logic goes so that you don't have something at all, not even a piece of it. That "a piece of" is expressed by the partitive case (as the name implies), so Sinulla ei ole läppäriä.
The last step in our analysis is to transform the negative statement into a question. There are a couple of ways of doing it in English, either "Haven't you got a laptop?" or the one in the exercise. In Finnish you use the question ending -ko/kö for this type of questions (a yes/no question), which you usually attach to the verb, thus Eikö sinulla ole läppäriä?
Thanks for the response! I think I'm beginning to understand the positive/negative split, but there's still one other point that I'm unclear about:
Take the sentence you've used, "Sinulla on läppäri". A natural translation into English might be "You have a laptop (computer)". But an extremely literal translation would probably be more like "By/at/on you is a laptop". "Laptop" is the subject of this English sentence, even though it comes last. "You" is the object of a preposition that isn't present as a standalone word in the Finnish.
How is "läppäri" not the subject of what looks like as close a translation as you can get in the Finnish sentence?