You may attach the patch after the mikveh, because the attachment is temporary in nature. We follow the view that, as with a band-aid, one should prepare the adhesive patch before Shabbat by removing and replacing the sticker-backings.
You should follow the patch booklet’s instructions. We do not know if the switch will increase the risk of breakthrough bleeding. You may wish to consult with your physician.
The halachic status of hormonally controlled uterine bleeding, such as that which occurs when using the patch, is a matter of debate among current authorities. One school of thought is to ignore the fact that you are using the patch and to keep the usual vestot (generally the Hebrew date, the interval and the thirtieth day). Another approach is to consider only the patch and to keep a veset of 2-4 days after cessation of the pills for that month. A third approach is to initially ignore the patch; however, if you establish a pattern of three months of bleeding at a consistent interval after cessation of the patch, then you establish a veset based on the interval from the cessation rather than the interval from the first day of bleeding. You should consult your rabbi as to how to proceed in practice. We follow the third approach. This is due to the fact that while most women will bleed 2-4 days after cessation of the hormones, this can vary from woman to woman.