I'm kinda confused - do you want to lose 20 lbs in 4 weeks or 20 lbs altogether and just want to make a good go of it within 4 weeks?
I always recommend this for a guide to food:
http://www.choosemyplate.gov/myplate/index.aspx No carbs are not the devil if you eat the right kind of carbs!
Fitness DVDs are great, doing weights will help with your arms. Jillian Michaels fitness DVDs tend to focus on weights so you might want to check her out if you haven't already.
If you want to do a crash diet (and they're called that for a reason!) then you probably will lose a lot of weight. But it's damaging your health all the sake of vanity. If you don't look great in a bikini this time, so what? You will one day if you lose weight the healthy way. Don't be so flustered over being in a bikini. Wear a tankini if you're paranoid, they can look just as good as bikinis and cover up any areas you don't want hanging out.
Based on your height and your weight you don't seem to be overweight at all. Your BMI suggests that you're on the upper end of a healthy weight. I'm overweight and I would wear a bikini - sure I don't look thin and healthy but it's not a big deal. Just be confident with what you've got and it won't matter what your weight is.
But a bikini body is made in the kitchen complimented by exercise. Don't think you can just crunch your way to a six pack - you need to get a balanced diet - not just eat cereal twice a day!
Hope this helps.