Regarding the type rating, as Flydubai is a UAE operater flying A6- registered aircraft, you would be issued with a UAE ATPL, endorsed with a B737 300-900 rating.
If you wanted the rating on your EASA licence as well you would need to find a friendly EASA TRE to give you a skills test and have it added on. I'd assume the process for an FAA licence is similar.