In order to benefit fully from your Monthly Card, you must log in every day. If you miss a day, you will not be able to retro-actively claim that Dilithium, and it will not bump forward the end date of the overall Monthly Card. You can claim your daily Dilithium by entering the Daily Missions menu at the bottom left of the main screen.
There are no automatically-renewing purchases in Star Trek Timelines. When a Monthly Card is purchased, it is valid for 30 calendar days, after which it would need to be re-purchased. If you have no desire to renew it next month, you need only let it run its course