You cannot drop her off for a day in an Assisted Living Facility, no. You'd need to find an adult daycare for errands & shopping, or hire someone to come into your home to care for your mom for a day out.
Respite care for vacations. Many AL facilities offer this, including for their MC, if they have space, but you will need to have all your mom’s health info and vaccines, etc., current. Find out ahead of time what is required. For just a day it would be easier to have in-home care while you go out.
Good luck!
https://www.assistedliving.org/respite-care/