Before drinking the sotah water does the woman make a blessing? If so would it be shahakol or something else?
You only recite a bracha on water if you are thirsty.1 Since the sotah is not drinking the water to quench her thirst, there would be no bracha. As for birkat hamitzvah I don't know.
The question has been raised in the comments about what if the sotah was thirsty when she drank the water. It seems (at least in the way I interpret it) that according to the Mishnah Berurah, she would recite a bracha.2
1 Shulchan Aruch, O.C. 204:7. Borei Nefashot would not be said as well. See also Brachot 44a and Rambam Brachot 8:1.
2 Mishnah Berurah 204:2. The case is drinking water in order to swallow a pill. If the person drinks the water in order to swallow the pill and to quench his thirst, according to this opinion he would recite a bracha on the water. Perhaps if the sotah was thirsty, she would recite a bracha as well.