To prevent underarm odor: rub 1/2 lime under your arm pits prior to bathing once a week (if you are a woman, make sure skin is not irritated from waxing/shaving). I also swear by WELEDA natural deodorant, comes in citrus and rose (as a woman, I prefer rose). It works wonders because it has natural citric oils, not synthetic.
For a white t-shirt you can try laundering with lime juice, but remember lime juice can whiten clothes if left on for a period of time, so do a test on an old shirt before proceeding! ... View related article.