Is it permissible to offer a votive fast that was restricted to a specific day, such as fasting on a Saturday, on the preceding Thursday, in the event that it is not possible to fulfill it on its designated day? And is it permissible to fast on a Saturday alone as part of a vow?

If you vowed to fast an unspecified Saturday, then you are required to fast any Saturday. However, if you vowed to fast a specific Saturday (such as a particular date), then you must fast that exact specified day, and fasting it earlier does not suffice. It is disliked to single out Saturday for voluntary fasting, but whoever vows to fast it is obligated to do so.