I was a Taiwan ETA a few years ago, and one of the other ETAs did this. Basically, Fulbright gives you an apartment and a nice stipend, the girl had to keep paying her rent for the Fulbright place (under 200 USD a month) but found another place for her and her boyfriend. Her boyfriend got a job teaching english and working in a kitchen. There are also opportunities, as an ETA, to put in some hours as a private tutor and make some more money on the side.
This was in Kaohsiung though, Yilan might be different (if you know where you're going yet). In general though, the Fulbright support staff is extremely helpful and accommodating there.