Keep yourself in optimum physical condition in the few days before the a cappella audition. Drink plenty of water. Avoid alcohol, which dehydrates the vocal chords. Avoid milk and orange juice, as they cause phlegm build up in the throat. In the days before the audition, get plenty of sleep--8 to 10 hours a night, if possible.
Understand that succeeding at an a cappella audition involves balancing a solo voice with the ability to blend. While classical training helps, a cappella groups are not necessarily looking for an operatically trained voice. While it is important to hold your own during a solo, a cappella groups like singers who will blend with the sound and harmony of the group. In a cappella groups, it is more important to be a team player than it is to be the best singer. If the group is going flat or sharp, you have to adjust to them, even when they are wrong. A cappella groups also like to have a pure sound. They like straight tone, as opposed to vibrato.
Know that ear training is important. At the audition, the group might play a progression of notes that you will have to sing back. Alternately, they may play a succession of chords and have you find and sing the middle note. Preparing for the ear-training part of the audition will require longer-term effort. While voice lessons will help your vocal technique, they may not necessarily help in ear training or musicianship. There are many ways to develop a good ear: Listen to different kinds of music, practice chords at the piano and take music-theory classes. Instrumentalists tend to have an advantage in the ear-training section of the audition. Instrumentalists are used to reading more complicated music and hearing different tonalities.
Recognize that a cappella groups also look for a personality fit. At the audition, they will ask you to prepare a joke. They want to see if you blend with the social dynamic of the group. A cappella groups like fun, outgoing people. Performing a cappella involves not only singing, but also stage presence, humor and the ability to perform off the cuff. That's why so many a cappella singers also have theatrical training.
Keep a level perspective. Sometimes, a rejection may have nothing to do with you. Maybe the group was looking for a different voice (an alto, whereas you are a soprano). Maybe the a cappella group had more of a pop sound, and you have an R&B voice. Maybe you didn't fit with the social dynamic of that particular group. A cappella groups do not necessarily look for the best singer. They look for the best fit for the group. If you keep getting rejected, work on your vocal technique, ear training and musicianship. If you are good, something will eventually work out.