The earlier you start the better in writing a Fulbright application. Think about what really interests you or excites you. Applicants with a passionate interest in a subject find it easier to write a proposal. Some countries have a language proficiency requirement, so be realistic. If you don't speak Siswati, then maybe research in Swaziland is not right for you.
Some countries are also much more popular than others and therefore are more competitive.
Take this into consideration when designing your study or research project. You should begin the application process in your Junior year by narrowing the focus of your study to a project that can be completed in one year. In one to two pages, the project statement should give the what, why, how, how long, and where of your specific project and should give the reader a succinct picture of what you want to accomplish. The curriculum vitae is a one page narrative that gives a picture of you as an individual. You may include special interests and abilities, career plans, life goals, etc.