Note: I am not a Muslim. While I do have a casual interest in religions, Islam is one is not one I’ve ever gotten around to digging into in detail, so this is entirely based on what I managed to find while searching around. I would strongly recommend checking all of this with someone in your community.
From what I can tell, the only restrictions in Ramadan are: fasting, no sexual behaviour, no fighting, lots of prayers (usually, although apparently not always, in a mosque - I get the feeling this is one of the things that varies as an individual). As a result, you shouldn’t need to worry about privacy any more during this time than otherwise.
If the prayers need to be done at a certain time, you could just look up ahead of time what time those prayers are, and either set alarms on your phone for those times, or just write them down and keep an eye on the clock. I don’t think you need an app to do this, but to each their own I suppose.
I know some Orthodox Jewish people avoid turning on lights etc during Shabbat as an extension of the ‘kindling flame’ rule or ‘doing commerce’ rule, I don’t know if Islam has a similar prohibition? If that’s the case, I would recommend just turning your phone off beforehand and putting it aside, but again, you don’t need an app to do that.
Good luck with your observances!