"Can eating crab make you sick?"
Yes it's very possible eating crab can make you sick. Some people even a small amount, can make them violently sick or even be deadly to ingest. However usually such people, are also sensitive or allergic to other types of shellfish. The only way to find out if this is the care for you, would be to get tested for it. Until such time though as you are tested for it, I strongly and seriously recommend NOT eating it, or any other shellfish. Otherwise in such a small amount, if you weren't sensitive/allergic to it, should have had no affect on you. People can go years without eating something, eat it with no real affect. Then there are those, who quite literally can eat something today, and be fine, eat it a week later, and become violently ill from it.
Edit point:
Not everyone goes into anaphylactic shock, eating shellfish in the beginning, if they've started to develop a problem with shellfish. With allergies, there's no real set pattern, at the onset of them. Which was why I said what I did in the manner I did.
Now as to the cousin eating it and being fine, that means the sushi itself was fine. It was your reaction to it. I will also agree on the point of it not being crab meat though, due to the expense. Also find out what was used in making the sushi beforehand. This can even possibly help to pinpoint the issue. So you may well be sensitive/allergic to fish and seafood overall. Again as I said before, go get tested for it, and the sooner the better.
Now there are people who have gone a lot longer without eating meaty, or fish/seafood, who have had no issues when they went and ate meat.