The project statement for an applicant to a program or course of study must present a compelling argument for the proposed program or coursework. Is it is unique in its focus or structure? Does it benefit from particular faculty or institutional resources? Simply preferring to study a particular subject in a particular place may not be enough. Don't hesitate to frame your argument for the program in light of your goals and future plans (academic and personal) - why is this particular program a lynchpin to your future?
Admission to the selected masters program is separate from the Fulbright application process and may run on a very different timeline. It is rare for an applicant to have an offer of admission at the Fulbright application deadline but important to demonstrate that you understand and meet the necessary qualifications for admission and that you have established contact with the program.
The letter(s) of support/affiliation for an applicant to a masters program or non-degree coursework at an institution abroad may be different from those for a research grant. It is important to have a letter from the program you hope to attend that verifies that you meet their requirements for admission and your application will be considered in the normal application cycle. If you propose to do research as part of your program, a letter from the faculty member who might oversee that research will be beneficial as well.
There are two important caveats for those applying to masters programs or to do coursework. The first is that even if you receive Fulbright funding, that does not guarantee admission to your program. If you are funded but not admitted, you must either turn down the grant or try to work something out with Fulbright to create a "plan B." Individual countries have absolute discretion over whether they will entertain a "plan B." The second is that Fulbright tuition benefits vary by country, with some covering or waiving all tuition, some providing partial subsidy, and some not paying any tuition costs. It is important to know this information up front so you can secure extra funding if needed.