If your lassi is ready, I suggest finishing it within the first hour for the best taste. You can store it in the fridge to make it last a couple of days, but finishing it on the same day is advisable.
If your concern is limited to aam panna, then store it in the fridge, and it should stay fresh for a couple of months; however, use it on an early basis. Do not store it in the fridge for a longer time.