Hi Paula,
You're right, when you make observations on the website, they will not show up on the Nature's Notebook app. Currently, the app only sends observations in one direction (from your phone to the database), it can't pull observations from the database. We are definitely aware of this limitation and will try to make improvements to this in the future!
As for an easy way to count how many days you observe each month, if you are an individual observer, observing at a Personal Site, the easiest way is to download your datafile via the link on the bottom right of your Observation Deck. Then you can count the dates in the file. You can also follow the link to Visualize My Data, where you could make a phenology calendar for one or more of your species to see your observations.
Please let me know if you have any other questions!